Systems and methods for modifying an ice-to-object interface

ABSTRACT

Systems and methods for thermally modifying an ice-to-object interface. One system includes a power supply configured to generate a magnitude of power. The magnitude of the power is sufficient to melt an interfacial layer of ice at the interface; typically the interfacial layer has a thickness in a range one micron to one millimeter. A controller may be used to limit the duration in which power supply generates the magnitude of the power, to limit unneeded heat energy dissipation into the environment. Modulating the pulsed heating energy to the interface modifies a coefficient of friction between the object and the ice.

RELATED APPLICATIONS

[0001] This application claims the benefit of U.S. provisionalapplication Serial No. 60/356,476, filed Feb. 11, 2002 and incorporatedherein by reference. This application also claims the benefit of U.S.provisional application Serial No. 60/398,004, filed Jul. 23, 2002 andincorporated herein by reference. This application also claims thebenefit of U.S. provisional application Serial No. 601404,872, filedAug. 21, 2002 and incorporated herein by reference.

BACKGROUND

[0002] Ice presents many problems to a variety of industries. An exampleof one such problem can be found in the aviation industry when ice formson surfaces of an aircraft. Ice on a surface of an aircraft, such as awing, can create hazardous conditions for the aircraft, particularlywhile the aircraft is in flight. Another example can be found in theground transportation industry when ice forms on a windshield of anautomobile. Ice on the windshield can create a hazardous drivingenvironment for the driver of the automobile. Removing the ice from suchsurfaces can minimize hazardous conditions.

[0003] Present systems for removing ice include electric heaters thatapply power to resistive elements to generate heat. Other presentsystems include chemical solutions that generate chemical reactions tothermally dissolve the ice. The electric heaters apply a magnitude ofpower to a resistive element to directly and proportionally melt all icefrom the surface in contact with the electric heaters. The chemicalsolutions may thermally dissolve the ice but do not last for extendedperiods of time and produce undesirable conditions for the naturalenvironment. These systems are inefficient since they seek to completelymelt all of the ice.

[0004] Methods to remove ice include using a mechanical scrapper.Mechanical scrappers are often used to address the problem of iceadhering to an object's surface. However, mechanical scrappers are oftenhand-held and unwieldy to operate. Furthermore, mechanical scrappers arenot always effective in removing ice and may damage the surface to whichthe ice has adhered.

[0005] Failure to properly remove ice from the surface of an object canhave potentially catastrophic results. For example, an overabundance ofice on an aircraft in flight can dangerously reduce lift force of theaircraft and deny proper operation of some aircraft components. Anotherexample includes a build up of ice on an automobile windshield; if theice is not removed, a driver's vision may become impaired to the pointthat the driver will no longer be able to properly navigate the vehicle.

SUMMARY OF THE INVENTION

[0006] The following patents and patent applications provides usefulbackground and are thus incorporated herein by reference: U.S. Pat. No.6,027,075; U.S. Pat. No. 6,427,946; PCT application PCT/US/25124, filedOct. 26, 1999; PCT application PCT/US/28330, filed Nov. 30, 1999; PCTapplication PCT/US/01858, filed Jan. 22, 2002; PCT ApplicationPCT/US00/35529, filed Dec. 28, 2000; U.S. patent application Ser. No.09/971,287, filed on Oct. 4, 2001; and U.S. patent application Ser. No.09/970,555, filed on Oct. 4, 2001.

[0007] In one aspect, a pulse de-icer system heats an interface to asurface of an object so as to disrupt adhesion of ice and/or snow (asused herein, ice and/or snow may sometimes be denoted as “ice”) with thesurface. To reduce the energy requirement, one embodiment of a pulsede-icer explores a very low speed of heat propagation in non-metallicsolid materials, including ice and snow, and applies heating power tothe interface for time sufficiently short for the heat to escape farfrom the interface zone; accordingly, most of the heat is used to heatand melt only very thin layer of ice (hereinafter “interfacial ice”).The system includes a power supply configured to generate a magnitude ofpower. In one aspect, the magnitude of the power has a substantiallyinverse-proportional relationship to a magnitude of energy used to meltice at the interface. The pulse de-icer system may also include acontroller configured to limit a duration in which the power supplygenerates the magnitude of the power. In one aspect, the duration has asubstantially inverse-proportional relationship to a square of themagnitude of the power. The power supply may further include a switchingpower supply capable of pulsing voltage. The pulsed voltage may besupplied by a storage device, such as a battery or a capacitor. Thebattery or capacitor can, thus, be used to supply power to a heatingelement that is in thermal communication with the interface. Optionally,the pulsed voltage may be directly applied to a heating element so as todisrupt the adhesion of ice at the surface. In another aspect, theheating element includes a thin film of conductive material or a thinfilm that includes a semiconductor material. The semiconductor materialdoes not preclude vision through the thin film, to facilitate use with acar windshield, for example as the “object.” The power supply maymodulate power to the semiconductor material to convert the power intothermal energy. The modulated power transfers an appropriate magnitudeof thermal energy that can disrupt the adhesion of the ice to thesurface.

[0008] In certain aspects, the capacitor is either a supercapacitor oran ultracapacitor. In certain other aspects, the power supply is aflywheel and/or a high voltage power supply. Power from the power supplycan be converted into thermal energy for disrupting the adhesion of iceto the surface of the object. For example, the system may use the powersupply to remove ice and snow from the surface of an aircraft, a tire,an automobile windshield, a boat, a road, a bridge, a sidewalk, afreezer, a refrigerator, a building, a runway, or a window. Thoseskilled in the art will understand that other objects may be de-icedwith a pulse de-icer system.

[0009] In another aspect, a thermal transfer system uses a heat storagesub-system connected with a heating element. The heating element mayinclude a thermally conductive material such as a metal. The heatingelement may further include a membrane attached to the heating element.The membrane is for example inflatable such that when the membrane isinflated, heat is deterred from transferring to the surface of theobject to be de-iced. As the membrane deflates, the heating elementtransfers thermal energy to the surface to disrupt the adhesion of iceto the surface. The membrane can be frequently inflated and deflated tomodulate the thermal energy transfer to the surface.

[0010] In another aspect of a thermal transfer system, the heatingelement includes two regions of thermally conductive material separatedby a thermal insulator. At least one of the regions of the thermallyconductive material is moveably attached to the thermal insulator suchthat when the regions are positioned in a particular way, the tworegions physically contact one another. The movement of at least one ofthe regions may be modulated at a certain frequency such that one regionof thermally conductive material transfers an appropriate magnitude ofthermal energy to the other region. The transfer of thermal energythereby disrupts the adhesion of ice to the surface of the other region.

[0011] In one aspect, a method is provided to thermally modifyinterfacial ice at the interface between an object and ice. The methodincludes the step of applying heating energy to the interface to melt aninterfacial layer of ice. The step of applying is then limited induration so that heating energy applied to the interface has a heatdiffusion distance within the ice that extends no more than through thethickness of the interfacial layer of ice.

[0012] In one aspect, the step of applying heating energy includes thestep of applying power at the interface with a magnitude that is atleast about inverse proportional to a magnitude of energy used to meltthe interfacial layer of ice. In a related aspect, the step of limitingduration includes the step of limiting duration of the step of applyingpower at the interface such that the duration is at least about inverseproportional to a square of the magnitude of the power.

[0013] In one aspect, the step of applying heating energy includes thestep of applying power to the interface with a magnitude that issubstantially inverse proportion to a magnitude of energy used to meltthe interfacial ice, and the step of limiting duration includes the stepof limiting the duration so that the duration is substantially inverseproportion to a square of the magnitude of the power.

[0014] In one aspect, the method includes the further step offacilitating refreezing of the interfacial layer of the ice to affect acoefficient of friction between the object and the ice. By way ofexample, the step of facilitating may include one or more of thefollowing steps: (1) waiting for refreezing after the step of limitingduration; (2) blowing cold air at the interface; and (3) misting waterat the interface.

[0015] In certain aspects herein, the object is one of an aircraftstructure, a windshield, a mirror, a headlight, a power line, a ski liftstructure, a rotor surface of a windmill, a rotor surface of ahelicopter, a roof, a deck, a building structure, a road, a bridgestructure, a freezer structure, an antenna, a satellite, a railroadstructure, a tunnel structure, a cable, a road sign, a snowshoe, a ski,a snowboard, a skate, and a shoe.

[0016] In another aspect, the step of applying heating energy to theinterface includes the step of applying heating energy to the interfaceto melt an interfacial layer of ice having a thickness that is less thanabout five centimeters. In one aspect, the method step limits theduration such that the interfacial layer of ice has a thickness that isless than about one millimeter. In a related aspect, heat diffusiondistance is further restricted by limiting pulse duration such that thethickness of interfacial ice is between about one micron and onemillimeter.

[0017] In one aspect, the step of limiting duration applies the heatingenergy to the interface for a maximum of 100 s. In another aspect, thestep of limiting duration limits duration of applied heat energy tobetween about 1 ms to 10 s.

[0018] In another aspect, the step of applying heating energy to theinterface includes the step of applying power to a heating element inthermal communication with the interface, within the object, and/or incontact with the interface. In a related aspect, the step of applyingheating energy may include the step of electrically resisting the powerwith the heating element.

[0019] In one aspect, the steps applying and limiting are repeated in aperiodic manner to generate a desired coefficient of friction betweenthe object and the ice.

[0020] In one aspect, power is reapplyied at the interface after theinterfacial layer refreezes to selectively control a coefficient offriction between the ice and the object while the object moves over theice.

[0021] Those skilled in the art appreciate that, in certain aspects, icemay include or be replaced by snow without departing from the scopehereby.

[0022] In one aspect, the object is a slider such as a shoe, asnowboard, or a ski.

[0023] A method is also provided for controlling a coefficient offriction between an object and ice, including the steps of:

[0024] (1) pulsing power to an interface between the object and the iceto melt an interfacial layer of ice at the interface and decrease thecoefficient of friction;

[0025] (2) facilitating refreezing of the interfacial ice at theinterface to increase the coefficient of friction; and

[0026] (3) repeating steps (1) and (2) in a controllable manner tocontrol an average coefficient of friction between the object and theice.

[0027] In one aspect, the step of facilitating refreezing includes thestep of moving the object over the ice to decrease temeprature of theobject. For example, a car tire may be heated and then rotated (duringcar motion) to put the heated tire in contact with an ice-covered road,to facilitate refreezing.

[0028] In one aspect, the step of pulsing power includes the steps ofblowing first air onto the object (e.g., a vehicle tire), the first airhaving a temperature above freezing, and moving the object in contactwith the ice. In a related aspect, the step of faciliting refreezingincludes the step of blowing second air onto the object (e.g., thetire), the second air having a temperature less than the temperature ofthe first air.

[0029] A slider is also provided, the slider having a surface intendedto interface with ice or snow. A power supply (e.g., a battery)generates power. A heating element is configured to convert the power toheat at the surface, the heat being sufficient to melt an interfaciallayer of ice at the interface. A controller controls delivery of powerto the heating element to control a coefficient of friction between theslider and the ice or snow.

[0030] By way of example, the slider may take the form a shoe, asnowboard, a ski, or a snowshoe.

[0031] In one aspect, the slider is in a form of a ski, a skate or asnowboard, and the controller is responsive to user commands to modulatepower applied to the surface such that speed of the slider iscontrollable. In this manner, for example, a skier may control her speeddown the ski slope, as desired.

[0032] In still another aspect, a windshield de-icer is provided. Thewindshield deicer has a windshield and a substantially transparentheating element disposed with the windshield that generates heat inresponse to applied power in a magnitude sufficient to melt aninterfacial layer of ice on the windshield.

[0033] In one aspect, the heating element is selected from visuallytransparent semiconductor material having an electron gap larger thanabout 3 eV. For example, the material may be one of ZnO, ZnS, andmixtures thereof.

[0034] In another aspect, the heating element is selected fromtransparent conductor material. For example, the transparent conductormaterial may be one of indium tin oxide (ITO), tin oxide, thin metalfilms, and mixtures thereof.

BRIEF DESCRIPTION OF THE DRAWINGS

[0035]FIG. 1 shows one pulse de-icer system for modifying an interfacebetween an object and ice;

[0036]FIG. 2 shows one pulse de-icer system;

[0037]FIG. 3 shows one pulse de-icer system;

[0038]FIG. 4 shows one pulse de-icer system;

[0039]FIG. 5 shows one pulse de-icer system;

[0040]FIG. 6 shows one pulse de-icer system as applied to an aircraftwing;

[0041]FIG. 7 shows one pulse de-icer heating element laminate;

[0042]FIG. 8 shows one pulse de-icer heating element;

[0043]FIGS. 9 and 10 illustrate an exemplary heat diffusion distanceover a given time for one pulse de-icer apparatus;

[0044]FIG. 11 shows a graph illustrating a dependence of de-icing timeand de-icing energy for one pulse de-icer system;

[0045]FIG. 12 shows one HF de-icer system for modifying an ice-to-objectinterface;

[0046]FIG. 13 shows one HF de-icer system;

[0047]FIG. 14 shows an analysis of one HF de-icer system;

[0048]FIG. 15 shows assembly views of one interdigitated circuit for usein one HF de-icer system;

[0049]FIG. 16 shows views of an exemplary interdigitated circuit for usein one HF de-icer system;

[0050]FIG. 17 shows a graph of frequency dependence of ice conductivityand ice dielectric permittivity;

[0051]FIG. 18 shows an exemplary circuit characterizing one HF de-icer;

[0052] FIGS. 19-29 graphically illustrate certain test analyses of thecircuit of FIG. 18;

[0053] FIGS. 30-35 show graphs illustrating one analysis of heattransfer through convection of one HF de-icer system and heat transferthrough a substrate of the HF de-icer system;

[0054]FIG. 36 shows one thermal transfer de-icer system for modifying anobject-to-ice interface;

[0055]FIG. 37 shows one thermal transfer de-icer system;

[0056]FIG. 38 shows one thermal transfer de-icer system;

[0057]FIG. 39 shows one pulse de-icer system, illustrating comparisonwith a thermal de-icer transfer system;

[0058]FIG. 40 shows one thermal transfer de-icer system;

[0059]FIG. 41 shows one thermal transfer de-icer system;

[0060] FIGS. 42-46 show graphs illustrating one analysis of a thermaltransfer de-icer system;

[0061]FIGS. 47 and 48 illustrate characteristics of one slider;

[0062]FIG. 49 shows one slider apparatus that illustrates testing offrictional changes at the ice-to-object interface;

[0063]FIGS. 50 and 51 illustrate an application of one slider in theform of a ski;

[0064]FIG. 52 illustrates one slider in the form of a snowboard;

[0065]FIG. 53 illustrates one slider in the form of a shoe;

[0066]FIG. 54 illustrates one slider in the form of a tire;

[0067]FIG. 55 illustrates a test configuration of one slider;

[0068]FIG. 56 illustrates one slider in the form of a track;

[0069]FIG. 57 illustrates one slider in the form of a ski;

[0070]FIG. 58 illustrates one slider in the form of a tire;

[0071]FIG. 59 illustrates a test configuration of one slider;

[0072]FIG. 60 shows a graph illustrating an exemplary relationshipbetween coefficients of friction of certain sliders and voltage appliedto heating elements affixed to the sliders;

[0073]FIG. 61 shows a graph illustrating an exemplary relationshipbetween static force of certain sliders and normal pressure of thesliders exerted on snow;

[0074]FIG. 62 shows a graph illustrating an exemplary relationshipbetween coefficients of friction of certain sliders and the voltageapplied to an affixed heating element;

[0075]FIG. 63 shows a graph illustrating an exemplary relationshipbetween coefficients of friction of one slider and the time required tostop the slider;

[0076]FIG. 64 shows a graph illustrating another exemplary relationshipbetween coefficients of friction of one slider and voltage applied to anaffixed heating element;

[0077]FIGS. 65 and 66 show graphs illustrating thermal energy andcooling time of one slider;

[0078]FIG. 67 shows one analysis of one slider illustratingfriction-enhancement for an embodiment wherein the slider forms a tire;and

[0079]FIGS. 68 and 69 illustrate one frictional analysis between aslider and snow.

DETAILED DESCRIPTION OF THE DRAWINGS

[0080] Certain embodiments described below pertain to systems andmethods for modifying an interface between an object and ice. In oneembodiment, for example, a system applies energy to the interfacebetween ice (or snow) and a surface of an object to remove ice from thesurface, in order to “de-ice” the object. In another embodiment, forexample, a system modulates melting at an interfacial layer of ice at anice-object interface such that a melted interfacial layer quicklyrefreezes to modify the coefficient of friction between the objectsurface and ice.

[0081] Certain embodiments of de-icers or sliders utilize alternatingcurrent (AC) high frequency (HF) power sources, while other embodimentsof de-icers or sliders utilize direct current (DC) power sources and/orthermal energy transfer systems (e.g., heat storage system).

[0082] Certain sections below are categorized with the followingheadings: Pulse De-Icer Systems; Heating Elements As Used In PulseDe-Icer Systems; Pulse De-Icer System Analysis; HF De-Icer Systems;Interdigitated Circuit For Use In An HF De-Icer System; HF De-IcerSystem Analysis; Thermal Transfer De-Icer Systems; Thermal TransferDe-Icer System Analysis; Methods Of Coefficient Of FrictionManipulation; and Coefficient Of Friction Manipulation Analysis.

[0083] In certain sections describing pulse de-icer systems, forexample, certain embodiments describe operations of removing ice bymelting an interfacial layer of the ice adhering to a surface of anobject. Heating elements of certain pulse de-icer systems may also beused to melt the interfacial layer, such as through an electricalconnection to a DC or AC power supply. Certain other embodiments ofpulse de-icer systems modulate heating at the ice-to-object interfacesuch that the object refreezes (during a cycle of non-heating) and acoefficient of friction changes between the object and the ice. Certainpulse de-icers operate as or with a slider, as discussed hereinbelow.

[0084] In certain sections describing HF de-icer systems, for example,certain embodiments describe operations of removing ice by melting aninterfacial layer of the ice that adheres to a surface of an object.Interdigitated electrodes of certain HF de-icer systems may be used tomelt the interfacial layer and may be powered with an AC power supply,for example.

[0085] Certain other embodiments of the HF de-icer systems may be usedto modify a coefficient of friction between ice and a “slider.” As usedherein, a “slider” is an object that may interface with ice and/or snow;it may “slide” thereon due to interaction with the ice and/or snow andthe coefficient of friction between the slider and the ice and/or snow.Examples of sliders include, but are not limited to, tires; skis;snowboards; shoes; snowmobile tracks; sleds; aircraft landing gear, etcetera.

[0086] In certain sections describing thermal transfer de-icer systems,for example, certain embodiments are used to remove ice by melting aninterfacial layer of the ice adhering to a surface of an object. Thethermal transfer de-icer systems can be described to include heatstorage sub-systems which store thermal energy. The thermal energy inthese storage sub-systems may be transferred to a heating element thatis in thermal communication with the object-to-ice interface. Certainembodiments of thermal transfer de-icer systems thus store thermalenergy and transfer that energy to an object-to-ice interfaceselectively and/or in a controllable manner.

[0087] Certain other embodiments below describe systems that modify acoefficient of friction between ice and a slider by melting aninterfacial layer of the ice adjacent to the slider. Once melted, theinterfacial layer of ice refreezes to create a bond between the sliderand the ice. This bond acts as a “brake” which increases the coefficientof friction to the slider and the ice. Such systems then re-melt theinterfacial layer to break the bond, again modifying the coefficient offriction. This modulated interaction of freeing and refreezing at theobject-to-ice interface may control the coefficient of friction to adesired amount. This controlled coefficient of friction is for exampleuseful in devices such as cross-country skis, snow shoes, shoes, tires,snowboards, skates, and other devices which interact with ice and snow.

Pulse De-Icer Systems

[0088] Pulse de-icer systems are now described. The pulse de-icersystems may be used to remove ice from a surface of an object. Thefollowing systems may also be used to melt an interfacial layer of iceand/or to modify a coefficient of friction of an object-to-iceinterface, as described in more detail below.

[0089]FIG. 1 shows one pulse de-icer system 10 for modifying aninterface 15 between an object 16 and ice 11. System 10 includes powersupply 12, controller 14, and heating element 13. In one embodiment,power supply 12 is configured for generating power with a magnitude thatis substantially inversely proportional to a magnitude of energy used tomelt interfacial ice (hereinafter “interfacial ice”) at interface 15.Heating element 13 is coupled to power supply 12 to convert the powerinto heat at interface 15. Controller 14 is coupled to the power supply12 to limit a duration in which heating element 13 converts the powerinto heat. In one embodiment, the duration in which heating element 13converts the power into heat at interface 15 is substantially inverselyproportional to a square of the magnitude of the power.

[0090] More particularly, when a heating power density W (watt/m²) isapplied for time t to an interface between ice and a substrate, the heatpropagates in a distance l_(Di) in ice and in a distance l_(DS) in thesubstrate. The thickness of these heated layers and their respectiveheat capacities then determine how much heat is absorbed. If λ_(i) andλ_(S) are respective thermal conductivities of the ice and substrate,ρ_(I) and ρ_(S) are respective densities, and C_(i) and C_(S) are therespective specific heat capacities, then for a heat flux Q_(i) in iceand a heat flux Q_(S) in the substrate, one skilled in the art of heatexchange will then appreciate the following:

Q _(i) ≈C _(i) l _(Di)ρ_(i)(T _(m) −T)   (Eq. 0-1)

[0091] where T_(m)−T is the temperature change of the interface,

Q _(S) ≈C _(S) l _(DS)ρ_(S)(T _(m) −T)   (Eq. 0-2)

[0092] $\begin{matrix}{l_{Di} = \sqrt{\frac{\lambda_{i}t}{\rho_{i}C_{i}}}} & \left( {{{Eq}.\quad 0}\text{-}3} \right) \\{l_{DS} = \sqrt{\frac{\lambda_{S}t}{\rho_{S}C_{S}}}} & \left( {{{Eq}.\quad 0}\text{-}4} \right)\end{matrix}$

[0093] Solving Eq. (0-1)-Eq. (0-4) for the total amount of heat escapedfrom the interface, one can find: $\begin{matrix}{Q = {{{Q_{i} + Q_{S}} \approx {W \cdot t}} = {\frac{\left( {T_{m} - T} \right)^{2}}{W}\left\lbrack {\sqrt{\rho_{i}c_{i}\lambda_{i}} + \sqrt{\rho_{s}c_{s}\lambda_{s}}} \right\rbrack}}} & \left( {{{Eq}.\quad 0}\text{-}5} \right)\end{matrix}$

[0094] where W is density of heating power on the interface.

[0095] In one embodiment, therefore, the above algebraic analysisreturns an approximate result for power requirements within one pulsede-icer system and associated method. An accurate mathematicalconsideration solves a system of partial differential equations topredict, for a de-icing time t and de-icing energy Q, the followingexemplary embodiment:

[0096] In the example, controller 14 may control the time in which poweris delivered to heating element 13 according to the followingrelationship: $\begin{matrix}{{t = {\frac{{\pi \left( {T_{m} - T} \right)}^{2}}{4W^{2}}\left\lbrack {\sqrt{\lambda_{i}\rho_{i}c_{i}} + \sqrt{\lambda_{s}\rho_{s}c_{s}}} \right\rbrack}^{2}},} & \left( {{{Eq}.\quad 1}\text{-}1} \right)\end{matrix}$

[0097] where

[0098] T_(m) is an ice melting temperature, T is an ambient temperature,λ is a thermal conductivity coefficient, ρ is the material density, andC is the material heat capacity (subscript “i” denotes ice and/or snowand subscript “s” denotes substrate material) and W is a power persquare meter.

[0099] In the example, controller 14 also controls the magnitude ofpower that is delivered to heating element 13 such that energy Q atinterface 15 is substantially inversely proportional to the magnitude ofpower. In the example, controller 14 controls the magnitude of poweraccording the following relationship: $\begin{matrix}{Q = {{W \cdot t} = {{\frac{{\pi \left( {T_{m} - T} \right)}^{2}}{4W}\left\lbrack {\sqrt{\rho_{i}c_{i}\lambda_{i}} + \sqrt{\rho_{s}c_{s}\lambda_{s}}} \right\rbrack}^{2}.}}} & \left( {{{Eq}.\quad 1}\text{-}2} \right)\end{matrix}$

[0100] Accordingly, to reach a desired temperature (e.g., to melt ice atinterface 15) with less energy, one increases heating power W whileapplying the heating power over a shorter period of time. By way ofcomparison, the simplified analysis result of Eq. 0-5 differs from themore precise solution of Eq. 1-2 by a factor of π/4=0.785. Theseequations are particularly useful to describe short power pulses when aheat diffusion length is less then the target object thickness (e.g.,the thickness of interfacial ice within interface 15).

[0101] In one embodiment, a more accurate approximation is found byadding the energy used to melt a very thin layer of interfacial ice andto heat a thin heater of thickness d_(heater), Q_(min):

Q _(min) =l _(i) ·q _(i)·ρ_(i) +d _(heater) C _(heater)ρ_(heater)(T _(m)−T),   (Eq. 1-3)

[0102] where

[0103] l_(i) is melted layer thickness, ρ_(i) is ice density, q_(i) isice latent heat of fusion, and C_(heater) and ρ_(heater) are heaterspecific heat capacity and density, respectively. Accordingly, in theexample, controller 14 may control the magnitude of power according thefollowing relationship: $\begin{matrix}\begin{matrix}{Q = {{\frac{{\pi \left( {T_{m} - T} \right)}^{2}}{4W}\left\lbrack {\sqrt{\rho_{i}c_{i}\lambda_{i}} + \sqrt{\rho_{s}c_{s}\lambda_{s}}} \right\rbrack}^{2} +}} \\{{{d_{i} \cdot q_{i} \cdot \rho_{i}} + {d_{heater}C_{heater}{\rho_{heater}\left( {T_{m} - T} \right)}}}}\end{matrix} & \left( {{{Eq}.\quad 1}\text{-}4} \right)\end{matrix}$

[0104] The energy of Eq. 1-4 is given per square meter (J/m²).Convective heat exchange can also be added to Eq. 1-4; but that term isusually neglected due to very short heating-pulse duration. When thesubstrate and/or ice layer is thinner than the heat diffusion lengths(Eq. 0-3, Eq. 0-4, respectively), the energy is even less than that inEq. 1-4.

[0105] In illustrative operation, system 10 may for example be used withan automobile to remove ice 11 from a windshield (as object 16). In thisexample, heating element 13 is transparent and embedded in thewindshield 16, and power supply 12 and controller 14 cooperate toprovide power that is sufficient to melt interfacial ice at interface 15in accordance with Eqs. 1-1 and 1-2.

[0106] To further illustrate operation of system 10, consider theproperties of ice:

λ_(I)=2.2 W m ⁻¹ K ⁻¹, ρ_(i)=920 kg m ⁻³ , c _(i)=2 kJ kg ⁻¹ K ⁻¹ , q_(i)=333.5 kJ kg ⁻¹.   (Eq. 1-5)

[0107] The properties of a typical windshield (e.g., as the substrate)are:

λ_(s)≈1 W m ⁻¹ K ⁻¹, ρ_(s)≈3000 kg m ⁻³ , c _(s)≈1.54 kJ kg ⁻¹ K ⁻¹.  (Eq. 1-6)

[0108] According to Eq. 1-1, the time it takes to reach the ice meltingpoint (0° C.) starting at −10° C. and at a power rate of 100 kW/m² ist≈0.142 second with a glass or glass-like substrate 16. The correctionfrom Eq. 1-3 may add about 0.016 second to the duration, i.e. about 10%.Reducing the peak heating power by a factor of ten (e.g., from 100 kW/m²to 10 kW/m²) further increases this time by about two orders ofmagnitude. Comparatively, at −30° C., the total de-icing time at W=100 kW/m² can be as long as 1.42 second. A corresponding total de-icingenergy Q at W=100 kW/m² and −10° C. may thus be defined as:$\begin{matrix}{Q = {{100\quad {kW}\text{/}{m^{2} \cdot 0.158}\quad \sec} = {15.8\quad {\frac{kJoule}{m^{2}}.}}}} & \left( {{{Eq}.\quad 1}\text{-}7} \right)\end{matrix}$

[0109] At the same temperature of −10° C. and a lower power of W=10kW/m², however, the energy Q given by Eq. 1-4 is: $\begin{matrix}{Q = {144\quad k{\frac{Joule}{m^{2}}.}}} & \left( {{{Eq}.\quad 1}\text{-}8} \right)\end{matrix}$

[0110] This result is by almost one order of magnitude larger than atW=100 kwatt/m².

[0111] One advantage of the foregoing example is that a decreasedde-icing energy is used, as compared to prior art systems, by about oneorder of magnitude by increasing the power rate by about one order ofmagnitude and while shortening the time of applied power by about twoorders of magnitude. By limiting the time power is applied to interface15, the drain of heat energy into the environment and into bulk ice 11is limited. Instead, more energy remains conformed to interface 15 formelting interfacial ice as a result of shorter power pulses.

[0112]FIG. 2 shows one pulse de-icer system 20 in accord with oneembodiment. De-icer system 20 has a DC power supply 22, a chargecapacitor 26, a resistive heating element 28, and a switch 24. DC powersupply 22 is configured for supplying power to charge capacitor 26 whenswitch 24 is closed on node 23. Capacitor 26, when cooperatively coupledto resistive heating element 28 via node 25, is configured for supplyinga magnitude of power in accordance with the equations of FIG. 1. Switch24 is for example operatively controlled by a controller or amicroprocessor to pulse current from capacitor 26 into resistive heatingelement 28 as switch 24 closes on node 25, in accordance with Eq. 1-1 ofFIG. 1. In one example, DC power supply 22 charges capacitor 26 whenswitch 24 is closed on node 23. Once capacitor 26 is charged, switch 24opens and then closes on node 25 to discharge current into resistiveheating element 28. Resistive heating element 28 then generatessufficient heating power to melt an interfacial layer of ice at theobject interface (e.g., interface 15, FIG. 1). Depending on theapplication of pulse de-icer system 20, melting the interfacial layer isuseful to remove ice from a surface of an object, prevent its formationon the surface, and/or modify its adhesion strength and/or change acoefficient of friction between the ice or snow and the object.

[0113]FIG. 3 shows one pulse de-icer system 30 in accord with oneembodiment. Pulse de-icer system 30 includes a pair of power buses 32, aheating element 34, a capacitor 38, a switch 36, and a power supply 37.Pulse de-icer system 30 is configured for removing ice adjacent toelement 34 (e.g., element 34 is disposed with, within and/or on theobject to be de-iced). In the illustrated embodiment of FIG. 3,capacitor 38 is a supercapacitor having a storage capacity of about1000F and a potential of about 2.5V, such as a PC2500 supercapacitorproduced by Maxwell Technology. Also in this embodiment, heating element34 has a 50 μm sheet of stainless steel foil affixed to a 1 cm thickPlexiglas plate; and power supply 37 is a 2.5V DC power supply. Switch36 may operate as a high current mechanical switch to limit a durationin which power supply 37 applies power to heating element 34.Optionally, switch 36 operates as an electrical switch that receives acontrol from a controller, such as controller 14 of FIG. 1. Resistanceof heating element 34 is about 6 mΩ. With an initial power density ofabout 40 kW/m², a total stored energy of about 3.125 kJ, and a totalenergy density of about 83.33 kJ/m², pulse de-icer system 30 effectivelyde-ices about 2 cm of ice on about 375 cm² of surface area inapproximately one second at an ambient temperature of about −10° C.,using an energy density of about 40 kJ/m².

[0114] In another embodiment of pulse de-icer system 30, capacitor 38 isa car battery, such as an EverStart® car battery with a peak current ofabout 1000A and a potential of about 12V. Also in this embodiment,heating element 34 has a 100 μm sheet of stainless steel foil affixed toa 1 cm thick Plexiglas plate. Switch 36 may for example bestarter-solenoid switch. With an initial power density of about 25kW/m², pulse de-icer system 30 effectively de-ices about 2 cm of icegrown on about 375 cm² of surface area in approximately two seconds atan ambient temperature of about −10° C., using an energy density ofabout 50 kJ/m^(2.) In another embodiment, power supply 37 is a 2.5V DCpower supply that charges capacitor 38.

[0115]FIG. 4 shows one pulse de-icer system 40 in accord with oneembodiment. Pulse de-icer system 40 utilizes a DC power supply 42, acapacitor 45, a resistive heating element 46, a DC-to-DC converter 44,and a switch 48. DC power supply 42 is configured for supplying powervia DC-to-DC converter 44 to charge capacitor 45 when switch 48 isclosed on node 41. DC-to-DC converter 44 may be configured for “steppingup” the voltage from DC power supply 42. In one example, DC-to-DCconverter 44 has boost electronics that boost the power of DC powersupply 42. In one embodiment, capacitor 45 cooperatively couples toresistive heating element 46 via node 43 and is configured to supply amagnitude of power in accordance with the equations of FIG. 1. Switch 48is then operatively controlled by varying means, such as a controller ora microprocessor, to pulse current from capacitor 45 into resistiveheating element 46 as switch 48 closes on node 43, for example inaccordance with Eq. 1-1 of FIG. 1. In one example, DC power supply 42charges capacitor 45 when switch 48 is closed on node 41. Once capacitor45 is charged, switch 48 opens and then closes on node 43 to dischargecurrent into resistive heating element 46. Resistive heating element 46then generates sufficient heating power to melt an interfacial layer ofice. Depending on the application of pulse de-icer system 40, meltingthe interfacial layer of the ice is for example useful to remove icefrom a surface of an object, to prevent its formation on the surface,and/or to modify a coefficient of friction between the ice and theobject. Pulse de-icer system 40 is also useful when large power suppliesare not available or with objects having small surface area in contactwith snow, such as a shoe (e.g., shoe 684, FIG. 61). In one embodiment,pulse de-icer system 40 is used as a “pulse brake” described in moredetail below.

[0116]FIG. 5 shows one pulse de-icer system 50 in accord with oneembodiment. Pulse de-icer system 50 is configured for de-icing anobject. Pulse de-icer system 50 has a de-icer 62, a pair of power buses64, a thermocouple 63, a thermocouple module 52, an amplifier 54, abattery 58, a starter/solenoid 59, a capacitor 61, a solid-state relay(SSR) 60, and a computer system 57. De-icer 62 is coupled to power buses64 for receiving power from battery 58. Computer system 57 is coupled tode-icer 62 through thermocouple module 52 and amplifier 54 to receivetemperature information about de-icer 62 through thermocouple 63.Computer system 57 may include an analog to digital (A/D) converterboard 55 configured to receive the temperature information in an analogform and to convert the analog temperature information into a digitalformat for use by computer system 57. Computer system 57 also couples tode-icer 62 through SSR 60 to control the duration and magnitude of thepower applied to de-icer 62, for example in accordance with theequations of FIG. 1. In one example, computer system 57 operativelycontrols SSR 60 and starter solenoid 59 to apply power from battery 58to de-icer 62.

[0117] SSR 60 may be replaced with an inductor 68 and a switch 65.Starter-solenoid 59 may also include an inductor 67 and a switch 66.Computer system 57 may additionally include a transistor-transistorlogic (TTL) module 56 to send control information to SSR 60, such thatwhen inductor 68 receives a step input from TTL module 56, inductor 68closes switch 65. Once switch 65 closes, capacitor 61 discharges intoinductor 67 to close switch 66. Once switch 66 closes, battery 58delivers power to de-icer 62. In one embodiment, computer system 57decouples power from de-icer 62 when the temperature rises to apredetermined level, as determined by thermocouple 63. In one example,computer system 57 receives temperature information from thermocouple 63via thermocouple module 52 and amplifier 54. Thermocouple module 52relays the temperature information to computer system 57. Amplifier 54amplifies the temperature information such that A/D converter board 55digitizes the temperature information for computer system 57. Once thetemperature of de-icer 62 reaches a predetermined level sufficient tomelt an interfacial layer of ice, computer system 57 directs TTL module56 to open switch 65 via inductor 68. Since switch 65 is open whencomputer system 57 determines that power should be decoupled fromde-icer 62, capacitor 61 discharges and switch 66 opens because inductor67 no longer maintains a voltage. As such, inductor 67 begins to chargecapacitor 61.

[0118] In one embodiment, de-icer 62 is made of 50 μm thick stainlesssteel and attached to a leading edge of a small aerofoil (e.g., aforward exposed portion of an aircraft wing). In this embodiment, theaerofoil has a span of about 20 cm and thickness of about 5 cm andde-icer 62 has dimensions of about 20 cm×10 cm.

[0119] System 50 was tested as follows. De-icer 62 was formed into anaerofoil and placed in an icing wind tunnel; it was tested at an airspeed of about 142 km/h at about −10° C. with about 20 μm waterdroplets. Atmospheric ice formed on the aerofoil. After ice grew toabout 5 mm to 10 mm thickness, computer system 57 directed battery 58 toapply power to de-icer 62 in a pulsed manner, such as that described inFIG. 5. With a power density W of about 100 kW/m² and a power pulseduration t of about 0.3 second, de-icer 62 melts the interfacial layerof ice to the aerofoil such that the adhesion of the ice to the aerofoilsurface is substantially modified and/or broken. The ice thereafter isremovable from the aerofoil surface by air drag force. The pulseduration in this example is longer than in the example of the windshieldde-icer because of the larger heat capacity in the metal-foil heater.

[0120]FIG. 6 shows one pulse de-icer system 70 as applied to aircraftwing 80, in accord with one embodiment. Pulse de-icer system 70 has apower supply 74 and a controller 78. Power supply 74 is configured forgenerating power with a magnitude that is substantially inverselyproportional to a magnitude of energy used to melt an interfacial layerof ice at an interface 73. As shown, interface 73 is the surface ofaircraft wing 80 that is in contact with ice and/or snow. Pulse de-icersystem 70 also has a heating element 75 coupled to power supply 74 toconvert the power into heat at interface 73. System 70 has a controller78 coupled to power supply 74 to limit a duration in which heatingelement 75 converts the power into heat. The duration in which power isapplied is for example inversely proportional to a square of themagnitude of the power.

[0121] In one embodiment, system 70 also includes an ice detector 72 anda temperature sensor 76. Temperature sensor 76 is coupled to interface73 to detect a temperature at interface 73. Temperature sensor 76provides temperature information about interface 73 in the form of afeedback signal to controller 78. Controller 78 then processes thetemperature information to control the manner in which power is appliedto heating element 75 and/or interface 73.

[0122] Ice detector 72 is configured to detect a thickness of ice oninterface 73. Ice detector 72 may for example include a grid ofelectrodes that facilitate measurement of ice thickness. Since ice has aunique dielectric constant that differs from the dielectric constants ofwater and air, the presence and thickness of ice may be determined bymeasuring inter-electrode capacitance of ice detector 72. Ice detector72 relays information about the ice (e.g., ice presence and thickness)to controller 78. Controller 78 processes the information to determinewhen power should be applied to heating element 75. In one embodiment,when ice on aircraft wing 80 reaches a certain thickness, controller 78automatically determines that the ice is to be removed and operativelycontrols power supply 74 to apply power to heating element 75.

[0123] An example of the operative characteristics of system 70 is nowdescribed. Consider a de-icer environment in which the ambienttemperature T is about −10° C., air speed is about 320 km/hour, andthickness of aircraft wing 80 is about 10 cm, with a convective heatexchange coefficient h_(c) of about 1200 watt/K·m² (based onexperimental data).

[0124] By way of comparison, a prior art de-icer system would operate toapply power W to the surface of aircraft wing 80 to maintain thetemperature T_(m) at the surface of aircraft wing 80 above the freezingpoint of water (e.g., 0° C.), as in the following equation:

W=h _(c)(T _(m) −T)=12 kwatt/m ².   (Eq. 6-1)

[0125] Maintaining that power for a period of three minutes results in alarge amount of energy Q, as determined by the following equation:$\begin{matrix}{W = {{{12 \cdot 10^{3}}\quad {\frac{watt}{m^{2}} \cdot 180}\quad \sec} = {432 \cdot {\frac{kJoule}{m^{2}}.}}}} & \left( {{{Eq}.\quad 6}\text{-}2} \right)\end{matrix}$

[0126] Pulse de-icer system 70, on the other hand, distinguishes fromthe prior art de-icer system by, among other features, melting aninterfacial layer of the ice at interface as opposed to all of the ice.In one example, pulse de-icer system 70 cleans the aerofoil of ice usingonly 30 kJoule/m². With three minute intervals between pulses, pulsede-icer system 70 consumes a very low “mean” power of: $\begin{matrix}{W_{mean} = {\frac{30\quad {kJoule}}{180\quad {s \cdot m^{2}}} = {0.167\quad {\frac{kwatt}{m^{2}}.}}}} & \left( {{{Eq}.\quad 6}\text{-}3} \right)\end{matrix}$

[0127] Specifically, the result of Eq. 6-3 is only 1.4% of what a priorart electrothermal de-icer uses, per Eq. 6-2.

[0128] In one embodiment, pulse de-icer system 70 pulses energy toheating element 75 according to the equations of FIG. 1. Heating element75 may for example include a grid of electrodes to melt the interfaciallayer of ice at interface 73. When ice thickness reaches a certainpreset value (e.g., 3 mm), controller 78 directs power supply 74 todeliver a short pulse of power to heating element 75. The duration ofthe pulse depends upon the temperature as supplied by temperature sensor76, power as supplied by power supply 74, and physical properties of asubstrate material (e.g., the surface of aircraft wing 80 and/or heatingelement 75). For example, the pulse duration in which power is appliedmay follow Eq. 1-1 of FIG. 1.

[0129] In one embodiment, pulse de-icer system 70 employs a secondtemperature sensor (not shown) near heating element 75 to improve powercontrol. For example, once the interfacial temperature reaches apredetermined value as pulse power is applied, controller 78 may directpower supply 74 to decouple power from heating element 75, therebyconserving energy usage.

[0130] Experimentation with various heaters, such as a HFdielectric-loss heater and a DC heater, yields results that conform totheoretical predictions described above. In certain embodiments herein,when a de-icing area is too large for the power supply to simultaneouslyheat the entire area, de-icing may be performed section-by-section. Byway of example, an entire structure may be de-iced by sequentially byde-icing these sections. Air-drag forces associated with an aircraft mayadditionally remove ice from an aerofoil; however, as it takes time tokeep the most forward-advanced portion of aircraft wing 80 unfrozen(e.g., a parting strip), this may increase the average power shown inEq. 6-3. Other heaters may be used with pulse de-icer system 70 withoutdeparting from the scope hereof, such as the hot bleed-air heater foundin many aircraft.

Heating Elements as Used in Pulse De-Icer Systems

[0131] In certain of the following embodiments, heating elements as usedin various pulse de-icer systems are described. These heating elementsfor example receive power from a power supply, such as a DC powersupply, and then melt an interfacial layer of ice at a surface-to-iceinterface of an object. Once the interfacial layer of ice is melted, theice is for example removed or refrozen depending on the desiredapplication, such as those applications described in more detail below.

[0132]FIG. 7 shows exemplary pulse de-icer heating element laminate 90for removing ice from a structure 92, for example by applying power inaccordance with the equations of FIG. 1. Laminate 90 includes anelectrical and substrate thermal insulator 94, an electricallyconductive layer 96, and a protective layer 98. Layer 96 receives powerand converts that power into heat to remove and/or prevent ice formationon structure 92. Layer 96 is for example one of various heating elementsdescribed herein. In one embodiment, laminate 90 includes a plurality ofindividual components affixed to structure 92, thereby forming “cells”in which ice can be discretely removed (e.g., removed cell by cell, orsection by section).

[0133] In one embodiment, the deliverable power to laminate 90 is in arange of about$10\quad \frac{kW}{m^{2}}\quad {to}\quad 100\quad {\frac{kW}{m^{2}}.}$

[0134] Accordingly, a power supply selected to deliver such power shouldhave a capacity of about$10\frac{k\quad J}{m^{2}}\quad {to}\quad 100\quad \frac{k\quad J}{m^{2}}$

[0135] depending on the desired de-icing time and the outsidetemperature. Certain power supplies with these characteristics are inthe form of chemical batteries, such as car batteries, supercapacitors,ultracapacitors, electrolytic capacitors, flywheels coupled withgenerators, DC/DC and DC/AC invertors, and combinations thereof.

[0136] Modem chemical batteries are known for high density of storedelectric energy (e.g., about 60 kJ/kg for a lead battery). However,chemical batteries have a relatively low power density. For example, acar battery can deliver up to about 1000A at twelve volts for about tenseconds, corresponding to a power of about 12 kW. A typical car batteryhas a large capacity of about Q≈12V×100A×3600 sec=4.32·10⁶ J. Therefore,for use in pulse de-icer systems and methods, the car battery mayeffectively de-ice areas up to about 1.5 m², which is ideal forautomobile windshields.

[0137] Supercapacitors and ultracapacitors are known as good suppliesfor both peak power and peak capacity. Certain supercapacitors can store10 kJ/kg and can deliver 1.5 kW/kg of power (e.g., the PC2500supercapacitor by Maxwell Technology). As power supplies,supercapacitors may be well suited for use with laminate 90 in pulsede-icer systems.

[0138] A flywheel made of light composite materials and coupled with agenerator provides another energy storage. Certain flywheels can storeup to about 2 MJ/kg and, when coupled with a generator, can deliver apower density of about 100 kW/kg. As an example, a motor-generatorinitially operates as a motor spinning the flywheel to a high speed. Themotor uses a low-power source, such as 100 watt to 1000 watt source(e.g., a battery). When peak power is needed, the coils of themotor-generator are disconnected from the low-power source and connectedto a low-impedance load (e.g., electrically conductive layer 96),thereby inverting a kinetic energy stored in the flywheel into heat.

[0139] Certain applications of pulse method de-icers may usehigh-electric impedance heaters (e.g., a resistive heating element of anautomobile windshield de-icer) and, therefore, may need a high-voltagepower supply. For example, an automobile windshield de-icer may useabout 120 volts and up to 240 volts. This voltage exceeds an outputvoltage of a typical car battery (e.g., about 12 volts) and that of asupercapacitor (e.g., about 2.5 volts). Instead of employing a bank ofbatteries to increase the voltage, DC/AC invertors or step-up DC/DCconverters can be used to increase the voltage.

[0140] Thin electrical heating layers (e.g., electrically conductivelayer 96, FIG. 7) are useful in reducing energy requirements andde-icing thermal inertia. Examples of materials that may be used aslayer 96 are thin metal foils, such as stainless steel foil, titaniumfoil, copper foil, and aluminum foil. Sputtering metals, alloys,conductive metal oxides, conductive fibers (e.g., carbon fibers) andconductive paints may be used as well. A typical thickness of layer 96may be in a range of about 50 μm to 100 μm; however, other ranges, suchas that of about 10 nm to 1 mm, may also be used.

[0141] In one optional embodiment, protective layer 98 is configured toprotect layer 96 from harsh environments. For example, layer 98 protectslayer 96 from abrasion, erosion, high-speed impacts, and/or scratches.Protective layer 98 may be either dielectric or conductive and applieddirectly to layer 96. For example, layer 96 may have relatively goodthermal conductivity properties and relatively high mechanical strength.Certain examples of materials that may be used as protective layer 98include TiN, TiCN, tungsten carbide, WC, Al₂O₃, SiO₂, Cr, Ni, CrNi,TiO₂, and AlTiO. Protective layer 98 may be applied to layer 96 bysputtering, chemical vapor deposition (“CVD”), physical vapor deposition(“PVD”), and/or sol-gel methods (e.g., a colloidal suspension of silicaparticles that is gelled to form a solid). Sputtering, as known to thoseskilled in the art, may include placing a substrate in a vacuum chamber.A plasma generated by a passive source gas (e.g., Argon) generates anion bombardment directed towards a target on the substrate, therebycausing material of the substrate to be “sputtered”. The sputteredmaterial condenses on the chamber walls and the substrate. CVD and PVDtechniques are known to those skilled in the art.

[0142] Because energy requirements for pulse method de-icers can dependon substrate properties (e.g., {square root}{square root over(ρ_(s)c_(s)λ_(s))} of Eqs. 1-1, 1-2, 1-4 ), de-icing power can belowered for substrate materials of low density, low heat capacity,and/or low thermal conductivity. Many polymers have low(ρ_(s)c_(s)λ_(s)) product while metals have high (ρ_(s)c_(s)λ_(s))product. Solid foams also have low (ρ_(s)c_(s)λ_(s)) product. Glass hasa (ρ_(s)c_(s)λ_(s)) product that is higher than that of a typicalpolymer, but comparatively lower than that of metals. Depending on theapplication, substrate thermal insulator 94 can be about 100 nm to 1mm-thick, but is typically about 0.1 mm to 20 mm thick.

[0143]FIG. 8 shows one pulse de-icer heating element 100, in accord withone embodiment. Heating element 100 is configured for melting aninterfacial layer of ice on an object by receiving pulsed energy, suchas in accordance with the equations of FIG. 1. For example, power may beapplied to heating element 100 at terminals 101 and 102 such thatheating element 100 melts an interfacial layer of ice. A power supply,such as those described herein, may supply power to heating element 100to melt the interfacial layer of ice. Depending on the application ofheating element 100, melting the interfacial layer of the ice may beuseful to remove ice from a surface of an object, prevent its formationon the surface, and/or modify its adhesion strength and change acoefficient of friction between ice and the object. Element 100 may bedisposed at, in, or adjacent to the object surface to be de-iced, forexample.

Pulse De-Icer System Analysis

[0144] Certain operative characteristics of various pulse de-icersystems are next analyzed and described. In the following exemplaryanalyses, certain component values are illustrated to show how heat froma heating element diffuses into ice to remove the ice from an object.

[0145]FIG. 9 shows one pulse de-icer apparatus 120. Illustratively, ice124 adheres to a thermally conductive substrate 126 forming anice-object interface 122. A heating element such as described herein isdisposed with interface 122 (e.g., within substrate 126) to facilitatedelivery of pulsed energy to interface 122. Substrate 126 represents astructure such as an aircraft wing, car windshield, window, outsidemirror, headlight, rotor of a windmill, building, road structure,bridge, refrigerator, antenna, communication tower, train, railway,tunnel, road sign, power line, high tension wire, ski lift structure orski lift cable.

[0146]FIG. 10 illustratively shows heat diffusion distance over a giventime t (e.g., t₁ and t₂), through ice 124 and substrate 126, from atemperature T at the ice-object interface 122. X-axis 123 representsdistance perpendicular to interface 122, as shown in FIG. 9; and Y-axis125 represents temperature T. Each curve t₁ or t₂ represents time forheat diffusion distance into thermally conductive substrate 126 and ice124 on opposing sides of interface 122. As shown, the peak of each curvet₁ and t₂ is at a melting point temperature 127 on Y-axis 125, i.e., thetemperature sufficient to melt an interfacial layer of ice at interface122.

[0147] The two curves t₁ and t₂ depend on pulsed power that melts theinterfacial layer of ice. As shown, t₁ is less than t₂ and therebycorresponds to a higher rate of power. Since a pulsed amount of energyapplied under either curve t₁ and t₂ is sufficient to melt theinterfacial layer of ice at interface 122, it is preferable to applysuch pulsed energy in accordance with t₁, which utilizes a higher rateof power but overall less power as compared to t₂.

[0148] More particularly, consider the following equation for diffusiontime t over a length L coincident with X-axis 123: $\begin{matrix}{{t = \frac{L^{2}}{D}},} & \left( {{{Eq}.\quad 10}\text{-}1} \right)\end{matrix}$

[0149] where

[0150] D is a coefficient of heat diffusivity set forth by:$\begin{matrix}{{D = \frac{\lambda}{\rho \quad c}},} & \left( {{{Eq}.\quad 10}\text{-}2} \right)\end{matrix}$

[0151] where

[0152] λ is a thermal conductivity coefficient, ρ is the materialdensity,,and c is the material heat capacity. Pulses of shorter durationpower applied to interface 122, accordingly, heats thinner interfaciallayers of ice. By controlling heating power duration, it is betterfocused at interface 122, where needed. In one embodiment, the time tand energy Q applied to interface 122—to heat an interfacial layer ofice 124 from an ambient temperature T to a melting point temperature127—follow the equations discussed in connection with FIG. 1. Byemploying the equations of FIG. 1, energy is saved when de-icing withapparatus 120. Additionally, the time t between heating pulses may becontrolled such that the time t is defined by a rate of ice growth andtolerance to ice thickness. For example, when ice reaches a thickness ofabout 3 mm on an aircraft wing, a feedback mechanism enables apparatus120 to remove ice 124 such as discussed in connection with FIG. 6.

[0153]FIG. 11 shows a dependence of de-icing time and de-icing energy(e.g., thermal energy) on the density of heating power for one pulsede-icer system as applied to a car windshield, in accord with oneembodiment. For example, a 0.5 μm layer of conductive indium-tin oxide(ITO) coated on one side of the windshield made of glass and havingdimensions of about 10 cm×10 cm×5 mm may be used as a heating element inthe pulse de-icer system. When ice grows on the windshield with about 2cm thickness in an environment of about −10° C., pulses of about 60 HzAC power are applied to the heating element to heat an interfacial layerof the ice. Once the interfacial layer of ice is melted, the force ofgravity may remove the ice. The thermal energy Q needed to melt theinterfacial layer of ice may depend on the time and power density inwhich power is applied to the heating element. FIG. 11 illustrates sucha dependency where Y-axis 132 represents de-icing time and de-icingenergy, and X-axis 133 represents heating power rate W; the time isshown in seconds and the energy is shown in kjoule/m².

[0154] Two plots 130 and 131 substantially conform to theoreticalpredictions given in Eq. 1-4 of FIG. 1. For example, plots 130 and 131show that the de-icing time is inversely proportional to a square of theheating power rate W, while the thermal energy Q is approximatelyinversely proportional to a first power of the heating power rate W.Accordingly, such a pulse de-icer system reduces the magnitude ofaverage power delivered to the heating element to remove ice from, orprevent its formation on, an object.

HF De-Icer Systems

[0155] HF de-icer systems are now described. HF de-icer systems are forexample used to remove ice from a surface of an object. As above, HFde-icer systems may melt an interfacial layer of ice at an object-to-iceinterface such that the adhesion of ice to the surface is disrupted,modified, and/or broken. Once the adhesion of the ice is disrupted, theice may be removed from the surface, such as by the force of gravityand/or windshear.

[0156]FIG. 12 shows HF de-icer system 140 in accord with one embodiment.HF de-icer system 140 has bifilar wound coil 141 implanted onto adielectric substrate 142. Illustratively, ice and/or snow 143 is shownadhered to a surface 144 of dielectric substrate 142. Coil 141 may becoated with a dielectric layer to prevent mechanical and environmentaldegradation and/or to prevent an electric breakdown of air. Windings ofcoil 141 are spaced on dielectric substrate 142 by a distance D. Whenpower is applied to coil 142, for example in accordance with theequations of FIG. 1, HF de-icer system 140 disrupts or modifies adhesionof ice and/or snow 143 from surface 144. Exemplary operativecharacteristics of HF de-icer system 140 are now described.

[0157] Typical ice has a capacitance per square meter of:$\begin{matrix}{C_{i} \cong {\frac{1.2 \times 10^{- 11}}{D(m)}\frac{F}{m^{2}}}} & \left( {{{Eq}.\quad 12}\text{-}1} \right)\end{matrix}$

[0158] and a HF-conductance per square meter of: $\begin{matrix}{{G_{i} \cong {\frac{0.53 \cdot 10^{- 4}}{D(m)} \cdot {^{6670{({\frac{1}{273} - \frac{1}{T{(k)}}})}}\left( \frac{1}{{ohm} \cdot m^{2}} \right)}}},} & \left( {{{Eq}.\quad 12}\text{-}2} \right)\end{matrix}$

[0159] where

[0160] D is in meters and T in Kelvins. Electric breakdown of air occursat a voltage V_(B) of about:

V _(B)≈2.4×10⁶ D(m).   (Eq. 12-3)

[0161] As calculated at sea level, and using the air-breakdown electricfield of about 30 kV/cm, the root mean squared (rms) voltage V_(B) isabout:

V _(B)≈1.7×10⁶ D(m).   (Eq. 12-4)

[0162] As a matter of design preference, the maximum voltage isdetermined to be about 70% of V_(B) in (Eq. 10-4), for safetyconsiderations. Accordingly, V_(max) is determined to be:

Vmax=0.7·1.7×10⁶ D(m)≈1.2×10⁶ D(m).   (Eq. 12-5)

[0163] Combining Eqs. 12-2 and 12-5, the maximum heating power W_(max)is determined to be: $\begin{matrix}{{W_{\max}(T)} = {{G_{i}V_{\max}^{2}} = {0.763 \cdot 10^{8} \cdot {D(m)} \cdot {^{6670{({\frac{1}{273} - \frac{1}{T}})}}.}}}} & \left( {{{Eq}.\quad 12}\text{-}6} \right)\end{matrix}$

[0164] De-icing time of HF de-icer system 140 is heuristicallydetermined by applying “safe” voltages according to the followingequation: $\begin{matrix}{{W\left( {T,V} \right)} = {{W_{\max}(T)} \cdot {\left( \frac{V}{V_{\max}} \right)^{2}.}}} & \left( {{{Eq}.\quad 12}\text{-}7} \right)\end{matrix}$

[0165] Assuming 0.5 mm wires in coil 141 and a safe voltage of 600 voltsrms, de-icing time of HF de-icer system 140 is heuristically determinedto be about thirteen seconds, to melt an interfacial layer of ice 143 atsurface 144 at an ambient temperature of −30° C. Other de-icing timesare heuristically determined to be about 4.3 seconds at an ambienttemperature of −20° C. and about 1.2 seconds at an ambient temperatureof −10° C.

[0166] It has been found that typical ice growth rate does not exceed1.5 mm/min. Accordingly, if desirous to shed (e.g., de-ice) ice 143 fromsurface 144 about every three minutes, approximate average powers forde-icing can be determined to be: $\begin{matrix}{{1.75\quad {kW}\text{/}m^{2}\quad {at}}\quad - {30{^\circ}\quad {C.}}} & \left( {{{Eq}.\quad 12}\text{-}8} \right)\end{matrix}$

[0167] The power density used to keep a 0.2 inch wide parting strip freeof ice may be determined by adding the power density of an eight-inchwide protective band to each of the power densities of Eq. 10-8,assuming a 40 kW/m² typical power density. For example, a typical powerdensity for the 5 mm-wide parting strip with an 8-inch wide protectiveband is determined as follows:

W=40(kwatt/m2)·0.2 inch/8 inch=1 kwatt/m ²   (Eq. 12-9)

[0168] Accordingly, adding Eq. 10-9 to the power densities of Eq. 12-8yields the following results:

4.1 kW/m² at −30° C.   (Eq. 12-8)

[0169] The power density of HF de-icer system 140 at −30° C. (e.g., 4.1kW/m²) is therefore only about 10% that of a prior art DC heater.

[0170]FIG. 13 illustrates another HF de-icer system 150 in accord withone embodiment. HF de-icer system 150 has a plurality of electrodes 154implanted onto a dielectric substrate 152 in the form of aninterdigitated electronic circuit. HF de-icer system 150 removes ice 151from surface 156 by applying electrical power to electrodes 154 from HFAC power supply 155. HF de-icer system 150 has de-icing characteristicsin which the density of heating power substantially depends on circuitdimensions a and b, where a is a distance between electrodes 154 and bis an electrode width. In one embodiment, electrodes 154 are woven intoa mesh.

[0171] As electrical power is applied to electrodes 154, electric fieldlines 153 form about electrodes 154, as shown. In HF de-icer system 150,circuit conductance G is proportional to circuit capacitance C persquare meter caused by electric field lines 153 above dielectricsubstrate 152. For example, $\begin{matrix}{{{G\quad {{\alpha C}\left( \frac{C}{G} \right)}} = \left( \frac{ɛ\quad ɛ_{0}}{\sigma} \right)},} & \left( {{{Eq}.\quad 13}\text{-}1} \right)\end{matrix}$

[0172] where

[0173] ε_(o) is free space permittivity (e.g., ε_(o)=8.85·10⁻¹² F/m ), εis a relative permittivity of ice, and σ is a conductivity of ice.Assuming a=b, the following can be concluded: $\begin{matrix}{{C \propto G \propto {\frac{1}{l} \cdot \frac{b}{a} \cdot {ɛɛ}_{0}} \propto \frac{1}{l} \propto \frac{\sigma}{l}},} & \left( {{{Eq}.\quad 13}\text{-}2} \right)\end{matrix}$

[0174] where l is equal to a plus b, also known as the structure period.The mean electric field E is: $\begin{matrix}{{E \approx \frac{V}{l}},} & \left( {{{Eq}.\quad 13}\text{-}3} \right)\end{matrix}$

[0175] where V is the rms voltage applied to the circuit of HF de-icersystem 150. Accordingly, the heating power W per cubic meter is:$\begin{matrix}{W = {{GV}^{2} \propto \frac{\sigma \quad V^{2}}{l} \propto {\sigma \cdot l \cdot {E^{2}.}}}} & \left( {{{Eq}.\quad 13}\text{-}4} \right)\end{matrix}$

[0176] Thus, if maximum heating power W_(max) is limited in HF de-icersystem 150 by the maximum possible electric field E_(max) (e.g.,breakdown field), then W_(max) follows the equation: $\begin{matrix}{W_{\max} \propto {\sigma \cdot l \cdot {E_{\max}^{2}.}}} & \left( {{{Eq}.\quad 13}\text{-}5} \right)\end{matrix}$

[0177] In this embodiment, therefore, W_(max) increases linearly as lincreases. Additionally, the volume density W_(max)^(v)

[0178] of W_(max) does not depend on l because: $\begin{matrix}{W_{\max}^{v} = {\frac{W_{\max}}{l} \propto {\sigma \cdot E_{\max}^{2}}}} & \left( {{{Eq}.\quad 13}\text{-}6} \right)\end{matrix}$

[0179] Therefore, to keep W constant, E is decreased as f increases.Accordingly, E can be reduced such that there is no corona discharge(e.g., beneficial when using polymer substrates and electrodeinsulation).

[0180] Experimentally, HF de-icer system 150 was operated at −12° C.with various heating powers and voltages and with electrodes havingdimensions of a=b=75 μm (e.g., when coated with 5 μm of polyimide film,such as a Kapton® polyimid, “Kapton”). The following results wereobtained: $\begin{matrix}\left\{ {\begin{matrix}{{{W = {1\quad {kW}\text{/}m^{2}}},\quad {{{at}\quad V} = {80\quad V}}}\quad} \\{{W = {2\quad {kW}\text{/}m^{2}}},\quad {{{at}\quad V} = {120\quad V}}}\end{matrix}.} \right. & \left( {{{Eq}.\quad 13}\text{-}7} \right)\end{matrix}$

[0181] Imposing new dimensions a=b=500 μm (e.g., a structure period ofmm), the voltage that maintain the power grows as a square root of theratio of new and previous structure periods, resulting in the following:$\begin{matrix}\left\{ \begin{matrix}{V^{\prime} = {{{\sqrt{\frac{500\quad {µm}}{75\quad {µm}}} \cdot 80}\quad V} \approx {207\quad {V\left( {1\quad {kW}\text{/}m^{2}} \right)}}}} \\{V^{\prime} = {{{\sqrt{\frac{500\quad {µm}}{75\quad {µm}}} \cdot 120}\quad V} \approx {310\quad {V\left( {2\quad {kW}\text{/}m^{2}} \right)}}}}\end{matrix} \right. & \left( {{{Eq}.\quad 13}\text{-}8} \right)\end{matrix}$

[0182] One advantage of HF de-icer system 150 is that its circuit may befabricated without photolithography, even on curved surfaces. Theelectric field strength may also decrease at a rate substantially equalto an increase in l.

Interdigitated Circuit for Use in HF De-Icer System

[0183] The following shows embodiments and analyses of interdigitatedcircuits which may be used as heating elements in HF de-icer systems.The heating elements may be configured to receive HF-AC power from an ACpower supply and used to melt an interfacial layer of ice at asurface-to-ice interface of an object. Once the interfacial layer of iceis melted, the ice may be removed or refrozen depending on the desiredapplication, such as those described below in the below section entitled“Methods Of Coefficient Of Friction Manipulation.”

[0184]FIG. 14 shows an analysis of HF de-icer system 140 of FIG. 13 inaccord with one embodiment. In this analysis, an improved $\frac{a}{b}$

[0185] ratio is determined for a given l. For example, $\begin{matrix}{{G \propto {\frac{1}{l} \cdot G^{\prime}}},} & \left( {{{Eq}.\quad 14}\text{-}1} \right)\end{matrix}$

[0186] where

[0187] G′ is per cell conductance. As conductance is proportional tocapacitance, G′ is proportional to the per cell capacitance as follows:$\begin{matrix}{{G^{\prime} \propto {C^{\prime} \cdot} \propto \frac{Q}{V} \propto {\int_{\frac{a}{2}}^{({a + \frac{b}{2}})}\frac{{E_{y} \cdot d}\quad x}{V}}} = {{\frac{1}{2}{\ln \left( \frac{{2\quad a} + b}{2a} \right)}} = {\frac{1}{2}{{\ln \left( \frac{a + 1}{2\quad a} \right)}.}}}} & \left( {{{Eq}.\quad 14}\text{-}2} \right)\end{matrix}$

[0188] From Eq. 14-2, the heating power can be determined as follows:$\begin{matrix}{W \propto {GV}^{2} \propto {\frac{G^{\prime}}{l} \cdot V^{2}} \propto {\frac{V^{2}}{l} \cdot {\ln \left( \frac{a + l}{2a} \right)}}} & \left( {{{Eq}.\quad 14}\text{-}3} \right) \\{{W \propto {\frac{V^{2}}{l}{\ln \left( \frac{a + l}{2a} \right)}}} = {\frac{E^{2}}{l^{\max}} \cdot a^{2} \cdot {{\ln \left( \frac{a + l}{2a} \right)}.}}} & \left( {{{Eq}.\quad 14}\text{-}4} \right)\end{matrix}$

[0189] where

[0190] 0≦a≦l. As shown in the graph of FIG. 14, when E is maintained asa constant, the maximum heating power W_(max) is reached at point 159where $\frac{a}{l} \cong 0.576$

[0191] (e.g., an approximation a≈b is relatively good since l=a+b). Theheating power W, when a=b=0.5 l, is approximately 97% of the maximumheating power W_(max). The graph of FIG. 14 also illustrates ratios of10% and 90% at respective points 157 and 158 where the heating power Wis found to be 17% and 43% of the maximum heating power W_(max). Incontrast, when voltage is maintained as a constant, wider electrodes(e.g., dimension “b”) increase the amount of heating power.

[0192]FIG. 15 shows assembly views 160-163 of an exemplaryinterdigitated circuit in accord with one embodiment. The interdigitatedcircuit of FIG. 15 may be used in a de-icer system such as thosedescribed in the HF de-icer systems and the pulse de-icer systemsdescribed above. In view 160, the interdigitated circuit is initiallyassembled by hard-anodizing one side (e.g., “hard anodized layer 172”)of thick aluminum foil 171. Hard anodized aluminum foil 171/172 isphysically mounted to polymer substrate 174 with adhesive 173 in view161. Once hard anodized aluminum foil 171/172 is mounted to polymersubstrate 174, electrodes are formed by etching and/or patening aluminumfoil 171 from the overall structure as shown in view 162 (e.g., patenededges 175). Afterwards, the structure is bent or fitted into a desirableshape as a matter of design choice. The remaining exposed side ofaluminum foil 171 is hard anodized to encapsulate the formed electrodesand to cure cracks in hard anodized layer 172 that result from bending,as shown in view 163.

[0193] While views 160-163 show one method of forming an interdigitatedcircuit, other methods of forming the interdigitated circuit fall withinthe scope hereof. Examples of other methods include etching and/orpatening copper foil to form copper electrodes and mounting the copperelectrodes to a Kapton substrate. An example of a copper interdigitatedcircuit on a Kapton substrate is shown in FIG. 16.

[0194]FIG. 16 shows two views of an exemplary interdigitated circuit 180in accord with one embodiment. Interdigitated circuit 180 includescopper anode 181, interdigitated electrode 182, copper cathode 183, andKapton substrate 184. Interdigitated circuit 180 may be formed in amanner similar to that discussed in FIG. 15. View 185 shows an isometricview of interdigitated circuit 180, while view 186 shows an overheadview. As shown in view 186, the pitch of interdigitated circuit 180defines the distal spacing between electrodes of interdigitatedelectrode 182. The pitch of interdigitated circuit 180 may also definethe distal spacing between electrodes of copper anode 181. The shift ofinterdigitated circuit 180 defines the spacing between electrodes ofinterdigitated electrode 182 and electrodes of copper anode 181. Thewidth of the interdigitated circuit 180 defines the width dimension ofthe electrodes of anode 181. The width of interdigitated circuit 180 mayalso define the width dimension of the electrodes of interdigitatedelectrode 182.

[0195] Interdigitated circuit 180 may be employed to modify frictionbetween an object and ice and/or snow by applying electrical power tointerdigitated electrode 182. For example, DC electrical power may beapplied to interdigitated electrode 182 according to the equations ofFIG. 1. In another example, AC electrical power may be applied tointerdigitated electrode 182.

[0196] In one embodiment, interdigitated circuit 180 modifies acoefficient of friction of an object's surface-to-ice interface incooperation with natural friction change between an object and ice orsnow over temperature. For example, a steel object “slider” slides onice when at a velocity of 3.14 m/s, the friction coefficient of theslider on the ice drops from 0.025 at −15° C. to 0.01 at −1° C. Toincrease the temperature of ice that is in direct contact with theslider, interdigitated circuit 180 can either heat the ice directlyusing HF electric fields or heat a surface of the slider.

[0197] Interdigitated circuit 180 may be affixed at the surface of theslider that is typically in contact with ice and snow. Either AC or DCelectrical power may be applied to interdigitated circuit 180 to heatthe surface of the slider. For example, application of the electricalpower to the surface of the slider according to the equations of FIG. 1may heat the ice and/or the surface and change the coefficient offriction between the slider surface and the ice.

[0198] In one embodiment, HF AC electrical power is applied tointerdigitated circuit 180 so as to directly heat the ice. When HF poweris applied to the electrodes of interdigitated circuit 180, electricfield lines, such as electric field lines 153 of FIG. 13, penetrate intoan interfacial layer of ice and generate Joule's electric heating in theice, as follows:

W _(h)=σ_(i) ·E ²,   (Eq. 16-1)

[0199] where

[0200] W_(h) is heating power in watts per cubic meter, σ_(i) isconductivity of ice or snow, and E is electric-field strength. Theelectric field penetrates ice or snow to a depth that is approximatelythe same as the distance d, or pitch, between the electrodes ofinterdigitated circuit 180. Accordingly, the heating power W_(h) followsthe equation: $\begin{matrix}{{W_{h} \approx {\sigma_{i} \cdot \frac{V^{2}}{d^{2}}}},} & \left( {{{Eq}.\quad 16}\text{-}2} \right)\end{matrix}$

[0201] where

[0202] V is the rms AC voltage. While the power W_(h) of Eq. 16-2relates to electric power per unit volume, power per square meter W_(s)of an ice/slider interface is of greater concern. To estimate the powerper square meter W_(s), the power W_(h) is multiplied by the thicknessof the heated layer, approximately d, as previously indicated.Therefore, the power per square meter W_(s) follows the equation:$\begin{matrix}{W_{s} \approx {\sigma_{i} \cdot {\frac{V_{2}}{d}.}}} & \left( {{{Eq}.\quad 16}\text{-}3} \right)\end{matrix}$

[0203] The heating power per square meter W_(s) may be limited by airelectric breakdown of electrified strength E_(b), therefore:$\begin{matrix}{\frac{V}{d} \leq E_{b} \approx {{3 \cdot 10^{6\quad}}V\text{/}m}} & \left( {{{Eq}.\quad 16}\text{-}4} \right)\end{matrix}$

[0204] From Eq's 16-3 and 16-4, the relation for maximum heating powerof HF voltage as measured per unit area of a slider is derived asfollows: $\begin{matrix}{W_{s} \leq {\sigma_{i} \cdot d \cdot {E_{b}^{2}.}}} & \left( {{{Eq}.\quad 16}\text{-}5} \right)\end{matrix}$

[0205] For substantially pure ice at −10° C., conductivity of the ice athigh-frequencies (e.g., greater than 10 kHz) is about 2·10⁻⁵ S/m.Inputting the values of conductivity σ_(i), electrified strength E_(b),and a distance of d≈0.25 mm (e.g., a typical dimension withinHF-deicers) into Eq. 16-5 establishes a maximum limit for HF-heatingpower at:

W _(s)≦45 kW/m ².   (Eq. 16-6)

[0206] A more realistic power used to increase the temperature of theinterfacial layer of ice by ΔT can be calculated according to thefollowing equation:

W _(speed) =l _(D) ·a·ν·ρ·C·ΔT,   (Eq. 16-7)

[0207] where

[0208] where ν is slider velocity, ρ is density of ice or snow, a isslider width, C is ice specific heat capacitance, and l_(D) is a heatdiffusion length in ice or snow. The heat diffusion length l_(D) is ofthe form:

l _(D) ={square root}{square root over (D·t)},   (Eq. 16-8)

[0209] where

[0210] t is time in which a particular location of ice is in contactwith the slider of the following form: $\begin{matrix}{{t = \frac{L}{v}},} & \left( {{{Eq}.\quad 16}\text{-}9} \right)\end{matrix}$

[0211] where

[0212] L is slider length, and D is a heat diffusion coefficient of thefollowing form: $\begin{matrix}{{D = \frac{\lambda}{C \cdot \rho}},} & \left( {{{Eq}.\quad 16}\text{-}10} \right)\end{matrix}$

[0213] where

[0214] λ is the thermal conductivity of ice or snow. Substitution ofEqs. 16-8, 16-9 and 16-10 into Eq. 16-7 yields the following powerestimate for modifying the coefficient of friction between the ice andthe slider:

W _(speed) =a·ΔT{square root}{square root over (ν·λ·C·L·ρ)}.   (Eq.16-11)

[0215] As a practical numerical example, two skis with a total width ofapproximately a=10⁻¹ m and a length of L=1.5 m may employ interdigitatedcircuit 180 to modify the coefficient of friction between the skis andsnow. Assume the skis are traveling at velocity of ν=10 m/s. Snowdensity ρ is $\begin{matrix}{{\rho = {{3 \cdot 10^{2}}\frac{\quad {kg}}{m^{3}}}},} & \left( {{{Eq}.\quad 16}\text{-}12} \right)\end{matrix}$

[0216] the change in temperature of the interfacial layer of snow ΔT is

ΔT=1° C.,   (Eq. 16-13)

[0217] and

[0218] the specific heat capacitance of snow C is $\begin{matrix}{C = {{2 \cdot 10^{3}}{\frac{J}{m \cdot K}.}}} & \left( {{{Eq}.\quad 16}\text{-}14} \right)\end{matrix}$

[0219] From these values, the power requirement estimate W_(speed) canbe calculated as follows:

W_(speed)=134 W.   (Eq. 16-15)

[0220] Since only a small fraction of the skis may actually be incontact with the snow at any given time, the power requirement estimateW_(speed) can be further decreased to a fraction of W_(speed), orW_(speed−fraction), according to the following: $\begin{matrix}{{W_{{speed}\text{-}{fraction}} = \frac{W}{H \cdot a \cdot L}},} & \text{(Eq.~~16-16)}\end{matrix}$

[0221] where

[0222] W is a skier's weight and H is a compressive strength of the snowin Pascals (Pa). For a heavy skier (e.g., 100 kg) and H=10⁵ Pa,W_(speed−fraction) can be calculated as:

W _(speed−fraction)≈6.6%.

[0223] Accordingly, the HF-power needed to modify the coefficient offriction is then:

W _(speed)=134 W×0.066≈9 W.   (Eq. 16-18)

[0224] While this embodiment shows one example of an application forinterdigitated circuit 180 (e.g., applied to skis), those skilled in theart should appreciate that interdigitated circuit 180 may be employed tomodify the coefficient of friction between ice and surfaces of otherobjects, including for example snowboards and snowshoes.

HF De-Icer System Analysis

[0225] Certain operative characteristics of various HF de-icer systemsare next analyzed and described. In the following exemplary analyses,certain component values are varied to illustrate various conditions,such as changing environmental conditions and/or changing heat transfermethods.

[0226]FIG. 17 shows a graph 190 illustrating frequency dependence of iceconductivity and ice dielectric permittivity. In graph 190, Y-axis 193represents permitivity ε and X-axis 194 represents frequency. Graph 190also summarizes HF heating power for interdigitated circuits, such asinterdigitated circuit 180 of FIG. 16.

[0227] When an electrically conductive material is placed in an electricfield E, a heat density per cubic meter W is generated as follows:

W=σE ²,   (Eq. 17-1)

[0228] where

[0229] σ is the material electrical conductivity (e.g., iceconductivity). As evident from Eq. 17-1, the heat density is linearlyproportional to the conductivity and is quadratically dependent on theelectric field strength. Therefore, to increase a heating rate and,thereby, reduce de-icing time, ice conductivity and/or electric fieldstrength may be increased.

[0230] Ice electrical conductivity depends on temperature, frequency,and impurities within the ice. Ice conductivity is illustrativelyincreased by adjusting a frequency of AC power used to modify acoefficient of friction between ice and a surface of an object. As such,frequency dependence of ice conductivity may be written as:$\begin{matrix}{{{\sigma^{\prime}(\omega)} = {\sigma_{s} + \frac{\omega^{2}{\tau_{D}^{2}\left( {\sigma_{\infty} - \sigma_{s}} \right)}}{1 + {\omega^{2}\tau_{D}^{2}}}}},} & \text{(Eq.~~17-2)}\end{matrix}$

[0231] where

[0232] σ_(s) and σ_(∞) are static and HF conductivities of ice,respectively, Ω is the radial frequency of the AC power, and τ_(D) is isan ice dielectric relaxation time.

[0233] In graph 190, conductivity varies as frequency is increased in anexemplary temperature environment of about −10.1° C. For example,conductivity increases with increasing frequency in curve 191 whileconductivity decreases with increasing frequency in curve 192.Accordingly, curves 191 and 192 illustrate different ways in which tovary the conductivity of an ice-object interface by adjusting HF heatingpower frequency.

[0234] In graph 190, at −10.1° C., ice has an electrical conductivity ofabout 0.1 μS/m at approximately 10 kHz. Ice conductivity decaysexponentially when temperature decreases. Accordingly, the conductivityof ice at −30° C. would be about one order of magnitude less thanconductivity of ice at −10° C.

[0235] Dimensions of an HF deicer heating element, such asinterdigitated circuit 180 of FIG. 16, may depend on ice conductivityand a desired rate of heating. Accordingly, when generating heat persquare meter W′ in a thickness of an interfacial layer of ice using anapplied voltage V with a distance d between the electrodes, the electricfield strength E follows the equation:

E=V/d.   (Eq. 17-3)

[0236] The heat per square meter W′, thereby follows the equation:

W′=W·d.   (Eq. 17-4)

[0237] After combining Eqs. 17-1 through 17-4, the heating power persquare meter is derived as follows:

W′=σV ² /d.   (Eq. 17-5)

[0238] As an example, a typical heating density for a car windshield isabout 1 kW/m² and a typical applied voltage V is about 100 volts. Usingthese values and the value for ice conductivity in Eq. 17-5 returns avalue of about 0.1 mm for the pitch of the electrodes. While thisexample provides typical estimates for electrode pitch, otherembodiments may vary. For example, ice conductivity and electrodedimensions may also depend on thickness and electrical properties ofprotective layers that coat the electrodes.

[0239]FIG. 18 shows an exemplary circuit 200 characterizing an HFde-icer in accord with one embodiment. Circuit 200 has an AC powersupply 201, a capacitor 203, a capacitor 204, a resistor 202, and aresistor 205. Resistor 202 is coupled to power supply 201 and tocapacitor 203 and has a resistance R_(s) representing an internalresistance of power supply 201. Resistor 205 is coupled in parallel withcapacitor 204 and has a resistance R_(i) representing ice resistance.Capacitor 204 has a capacitance C_(i) representing an ice layercapacitance. Capacitor 203 is coupled to resistor 205 and capacitor 204and has a capacitance C_(d) representing capacitance of a protectivedielectric layer on de-icing electrodes, such as coil 141 shown anddescribed in FIG. 12. Circuit 200 represents an electric circuit diagramsuitable to simulate and analyze certain de-icing systems hereof.

[0240] FIGS. 19-23 graphically illustrate certain test analyses ofcircuit 200 in accord with one embodiment in which circuit 200 has adielectric layer that envelops electrodes (e.g., a circuit such asinterdigitated circuit 180, FIG. 16, with a dielectric layer envelopingthe electrodes). In this embodiment, circuit 200 may be characterized bythe following Table 19-1: TABLE 19-1 ε₀ := 8.85 · 10⁻¹² f := 10, 100 . .. 1 · 10⁵ ω(f) := 2 · π · f T := 243, 244 . . . 273${\tau_{D}(T)}:={1.5 \cdot 10^{- 4} \cdot {\exp \quad\left\lbrack {6670\left( {\frac{1}{T} - \frac{1}{253}} \right)} \right\rbrack}}$

${ɛ_{s}(T)}:=\frac{25047}{T}$

ε_(inf) := 3.2${\sigma_{\inf}(T)}:={1.8 \cdot 10^{- 5} \cdot {\exp \quad\left\lbrack {6670\left( {\frac{1}{253} - \frac{1}{T}} \right)} \right\rbrack}}$

σ₀ := 10⁻⁸${ɛ\left( {f,T} \right)}:={ɛ_{\inf} + \frac{\left( {{ɛ_{s}(T)} - ɛ_{\inf}} \right)}{1 + \left( {{\tau_{D}(T)} \cdot {\omega (f)}} \right)^{2}}}$

${\sigma \left( {f,T} \right)}:={\left\lbrack \frac{\left\lceil {\left( {{\sigma_{\inf}(T)} - \sigma_{0}} \right) \cdot \left( {{\tau_{D}(T)} \cdot {\omega (f)}} \right)^{2}} \right\rceil}{1 + \left( {{\tau_{D}(T)} \cdot {\omega (f)}} \right)^{2}} \right\rbrack + \sigma_{0}}$

d := 10⁻⁷, 2 · 10⁻⁷ . . . 3 · 10⁻⁵ ε_(d) := 9.9${C_{d}(d)}:=\frac{ɛ_{0} \cdot ɛ_{d}}{8d}$

l := 2.5 · 10⁻⁴ V = 500 R_(S) := 0 $\begin{matrix}{{R_{i}\left( {f,T,d} \right)}:=\frac{4\left( {{\frac{3}{2}1} - {2 \cdot d}} \right)}{\sigma \left( {f,T} \right)}} \\{{C_{i}\left( {f,T,d} \right)}:=\frac{ɛ_{0} \cdot {ɛ\left( {f,T} \right)}}{4\left( {{\frac{3}{2}1} - {2 \cdot d}} \right)}}\end{matrix}\quad$

${Z_{i}\left( {f,T,d} \right)}:=\frac{R_{i}\left( {f,T,d} \right)}{2{\pi \cdot f \cdot i \cdot {C_{i}\left( {f,T,d} \right)} \cdot \left( {{R_{i}\left( {f,T,d} \right)} + \frac{1}{2 \cdot \pi \cdot f \cdot i \cdot {C_{i}\left( {f,T,d} \right)}}} \right)}}$

$\begin{matrix}{{Z\left( {f,T,d} \right)}:={{Z_{i}\left( {f,T,d} \right)} + \frac{1}{2 \cdot \pi \cdot f \cdot i \cdot {C_{d}(d)}}}} \\{{I\left( {f,T,d} \right)}:={\frac{V}{R_{S} + {Z\left( {f,T,d} \right)}}\pi}}\end{matrix}\quad$

P_(i)(f, T, d) := V · Re(I(f, T, d)) ε_(w) := 80 σ_(w) = 5 · 10⁻⁴${R_{w}(d)}:=\frac{4\left( {{\frac{3}{2}1} - {2 \cdot d}} \right)}{\sigma_{w}}$

${C_{w}(d)}:=\frac{ɛ_{0} \cdot ɛ_{w}}{4\left( {{\frac{3}{2}1} - {2 \cdot d}} \right)}$

${Z\left( {f,d} \right)}:=\frac{R_{w}(d)}{2{\pi \cdot f \cdot i \cdot {C_{w}(d)} \cdot \left( {{R_{w}(d)} + \frac{1}{2 \cdot \pi \cdot f \cdot i \cdot {C_{w}(d)}}} \right)}}$

${\begin{matrix}{{Z_{w}\left( {f,d} \right)}:={{Z\left( {f,d} \right)} + \frac{1}{2 \cdot \pi \cdot f \cdot i \cdot {C_{d}(d)}}}} \\{{I_{w}\left( {f,d} \right)}:=\frac{V}{R_{S} + {Z_{w}\left( {f,d} \right)}}}\end{matrix}\quad}\quad$

P_(w)(f, d) := V · Re(I_(w)(f, d))

[0241] where ε₀ is free space permittivity, f is incremental frequency,ω is radial frequency as a function of f, T is incremental ambienttemperature in K, τ_(D) is an ice dielectric relaxation time, ε_(s) is astatic dielectric permittivity of ice, ε_(inf) is a high-frequencypermittivity of ice, σ_(inf) is a high-frequency conductivity of ice, σ₀is a static conductivity of ice, ε is an ice permittivity (e.g., as afunction of frequency f and temperature T), σ is an ice conductivity(e.g., as a function of frequency f and temperature T), d is a thicknessof the protective dielectric layer, ε_(d) is a permittivity of theprotective dielectric layer l, V is voltage, Z_(i) is impedance of ice(e.g., as a function of frequency f, temperature T, and distance d),Z(f,T,d) is a total circuit impedance with ice covering the electrodes(e.g., as a function of frequency f, temperature T, and distance d), Iis applied current (e.g., as a function of frequency f, temperature T,and distance d), P_(i) is power delivered to heat the ice (e.g., as afunction of frequency f, temperature T, and distance d), ε_(w) is apermittivity for water, σ_(w) is a conductivity for water, R_(w) is awater resistance, C_(W) is a water capacitance, Z(T,d) is a totalcircuit impedance with water covering the electrodes (e.g., as afunction of frequency f and distance d), Z_(w) is impedance for water(e.g., as a function of frequency f and distance d), I_(w) is appliedcurrent (e.g., as a function of frequency f and distance d), and P_(w)is power delivered to the water (e.g., as a function of frequency f anddistance d). Electric power was calculated for both of the followingcases: when ice covers the electrodes, and when ice was melted and wateris in contact with the electrodes.

[0242]FIG. 19 illustrates the dependence of heating power generated indistilled water (i.e., plot 210) at 20° C. and in ice (i.e., plot 211)at −10° C. on a thickness of a dielectric coating on the electrodes. InFIG. 19, Y-axis 213 represents heating power per m² and X-axis 212represents thickness of the dielectric coating, in meters. In thisembodiment, the coating was an alumina coating. The frequency of the ACpower was about 20 kHz at a voltage of about 500 volts rms. At a coatingthickness of about 25 μm, the heating powers for water and ice areapproximately equal.

[0243]FIG. 20 illustrates the dependence of heating power generated indistilled water (i.e., plot 220) at 20° C. and in ice (i.e., plot 221)at −10° C. on frequency. In FIG. 20, Y-axis 223 represents heating powerin watt/m² and X-axis 222 represents frequency in Hz. At about afrequency of 20 kHz, the respective heating powers for water and ice areequal. It is useful to match the heating powers for water and ice toprevent cold or hot patches on the de-icer at which ice melted.

[0244]FIG. 21 illustrates the dependence of heating power generated inice (e.g., plot 230) on temperature. In FIG. 21, Y-axis 231 representsheating power in watt/m² and X-axis 232 represents temperature in K.Accordingly, a dielectric coating on the electrodes of an HF deicer maybe used to tune of de-icer performance.

[0245]FIG. 22 illustrates the dependence of a heat transfer coefficient(watt/m² K) on air velocity (m/s) (i.e., plot 240). In FIG. 22, Y-axis241 represents heat transfer coefficient h and X-axis 242 representsvelocity ν. FIG. 22 may assist in determining calculations of HF powerfor de-icing and/or anti-icing on a flat windshield. The size of awindshield used within FIG. 22 is 0.5 m. In the illustrated embodiment,circuit 200 operates as a HF de-icer with differing modes, such as ade-icing mode and an anti-icing mode, as applied to the windshield.Table 19-2 shows a MathCad file used to calculate the convective heatexchange coefficient for the car windshield: TABLE 19-2 v := 1, 1.1 . .. 30 L := 0.1, 0.2 . . . 1 Re_(tr) := 10⁵ ν := 1.42 · 10⁻⁵${{Re}_{L}\left( {v,L} \right)}:=\frac{v \cdot L}{\nu}$

k := 0.0235 Pr := 0.69 Re_(L)(20, 0.5) = 7.042 × 10⁵ $\begin{matrix}{{h\left( {v,L} \right)}:={\frac{k}{L} \cdot \left\lbrack {{0.664\quad {{Re}_{tr}^{0.5} \cdot \Pr^{\frac{1}{3}}}} + {0.036{{{Re}_{L}\left( {v,L} \right)}^{0.8} \cdot}}} \right.}} \\\left. {\Pr^{0.43} \cdot \left\lbrack {1 - \left( \frac{{Re}_{tr}}{{Re}_{L}\left( {v,L} \right)} \right)^{0.8}} \right\rbrack} \right\rbrack\end{matrix}\quad$

[0246] where

[0247] v is air velocity, L is a length of the windshield surface, Re isa range of Reynolds number from 10⁵ to 10⁷, h(v, L) is a heat transfercoefficient (e.g., as a function of voltage and L), k is on air thermalconductivity, and Pr is air Prandtl number, and ν is the air kinematicviscosity coefficient. In this embodiment, the heat transfer coefficienth(v, L) at about 30 m/s and a length of about 0.5 meters was 89.389w/m²K. Accordingly, FIG. 22 graphically shows (in plot 240) therelationship of the heat transfer coefficient h(v, L) to air velocity.

[0248]FIG. 23 illustrates one dependence of minimum HF power W_(min) ofcircuit 200 on outside temperature T (in °) for vehicle velocities of 10m/s (plot 252), 20 m/s (plot 251), and 30 m/s (plot 250). In FIG. 23,Y-axis 253 represents minimum HF power W_(min) (watt/m²) and X-axis 254represents temperature T. The minimum heating power W_(min) to maintainthe outer surface of the windshield at about 1° C. is shown in thefollowing Table 19-3 (MathCad file): TABLE 19-3 S:=0,0.1 . . . 2 T:=0,−1. . . −30 W_(min)(v,L,T,S):=h(v,L)·S·(1−T), where S is the windshieldarea

[0249] where

[0250] S is the windshield area.

[0251] Accordingly, plots 250, 251, and 252 may assist in makingdecisions with respect to applying power according to the velocity ν ofa vehicle using circuit 200.

[0252] FIGS. 24-26 graphically illustrate another analysis of circuit200, FIG. 18, in which circuit 200 has a dielectric layer that envelopselectrodes (e.g., a circuit such as interdigitated circuit 180, FIG. 16with a dielectric layer enveloping the electrodes). In this embodiment,circuit 200 may be characterized by the following Table 24-1 (MathCadfile): TABLE 24-1 ε₀ := 8.85 · 10⁻¹² f := 10, 100 . . . 1 · 10⁵ ω(f) :=2 · π · f T := 243, 244 . . . 273${\tau_{D}(T)}:={1.5 \cdot 10^{- 4} \cdot {\exp \left\lbrack {6670\left( {\frac{1}{T} - \frac{1}{253}} \right)} \right\rbrack}}$

${ɛ_{s}(T)}:=\frac{25047}{T}$

ε_(inf) := 3.2${\sigma_{\inf}(T)}:={1.8 \cdot 10^{- 5} \cdot {\exp \quad\left\lbrack {6670\left( {\frac{1}{253} - \frac{1}{T}} \right)} \right\rbrack}}$

${ɛ\left( {f,T} \right)}:={ɛ_{\inf} + \frac{\left( {{ɛ_{s}(T)} - ɛ_{\inf}} \right)}{1 + \left( {{\tau_{D}(T)} \cdot {\omega (f)}} \right)^{2}}}$

${\sigma \left( {f,T} \right)}:={\left\lbrack \frac{\left\lceil {\left( {{\sigma_{\inf}(T)} - \sigma_{0}} \right) \cdot \left( {{\tau_{D}(T)} \cdot {\omega (f)}} \right)^{2}} \right\rceil}{1 + \left( {{\tau_{D}(T)} \cdot {\omega (f)}} \right)^{2}} \right\rbrack + \sigma_{0}}$

d := 10⁻⁷, 2 · 10⁻⁷ . . . 3 · 10⁻⁵ ε_(d) := 9.9${C_{d}(d)}:=\frac{ɛ_{0} \cdot ɛ_{d}}{8d}$

l := 2.5 · 10⁻⁴ V = 500 R_(S) := 0 $\begin{matrix}{{R_{i}\left( {f,T,d} \right)}:=\frac{4\left( {{\frac{3}{2}1} - {2 \cdot d}} \right)}{\sigma \left( {f,T} \right)}} \\{{C_{i}\left( {f,T,d} \right)}:=\frac{ɛ_{0} \cdot {ɛ\left( {f,T} \right)}}{4\left( {{\frac{3}{2}1} - {2 \cdot d}} \right)}}\end{matrix}\quad$

${Z_{i}\left( {f,T,d} \right)}:=\frac{R_{i}\left( {f,T,d} \right)}{2{\pi \cdot f \cdot i \cdot {C_{i}\left( {f,T,d} \right)} \cdot \left( {{R_{i}\left( {f,T,d} \right)} + \frac{1}{2 \cdot \pi \cdot f \cdot i \cdot {C_{i}\left( {f,T,d} \right)}}} \right)}}$

$\begin{matrix}{{Z\left( {f,T,d} \right)}:={{Z_{i}\left( {f,T,d} \right)} + \frac{1}{2 \cdot \pi \cdot f \cdot i \cdot {C_{d}(d)}}}} \\{{I\left( {f,T,d} \right)}:={\frac{V}{R_{S} + {Z\left( {f,T,d} \right)}}\pi}}\end{matrix}\quad$

P_(i)(f, T, d) := V · Re(I(f, T, d)) ε_(w) := 80 σ_(w) = 5 · 10⁻⁴${R_{w}(d)}:=\frac{4\left( {{\frac{3}{2}1} - {2 \cdot d}} \right)}{\sigma_{w}}$

${C_{w}(d)}:=\frac{ɛ_{0} \cdot ɛ_{w}}{4\left( {{\frac{3}{2}1} - {2 \cdot d}} \right)}$

${Z\left( {f,d} \right)}:=\frac{R_{w}(d)}{2{\pi \cdot f \cdot i \cdot {C_{w}(d)} \cdot \left( {{R_{w}(d)} + \frac{1}{2 \cdot \pi \cdot f \cdot i \cdot {C_{w}(d)}}} \right)}}$

${\begin{matrix}{{Z_{w}\left( {f,d} \right)}:={{Z\left( {f,d} \right)} + \frac{1}{2 \cdot \pi \cdot f \cdot i \cdot {C_{d}(d)}}}} \\{{I_{w}\left( {f,d} \right)}:=\frac{V}{R_{S} + {Z_{w}\left( {f,d} \right)}}}\end{matrix}\quad}\quad$

P_(w)(f, d) := V · Re(I_(w)(f, d))

[0253] where the variables are the same as those found in Table 19-1,but with different values. For example, σ_(w) is the conductivity forwater with the same value of 5×10⁻⁴ S/m

[0254] FIGS. 24-26 graphically illustrate a dependence of heating powergenerated in distilled water (plots 261, 270, 281 of respective FIGS.24, 25 and 26) at 20° C. and in ice (plots 260, 271, 280 of respectiveFIGS. 24, 25 and 26) at −10° C., which differ in the thickness of thedielectric layer: 10⁻⁵ m (FIG. 24), 10⁻⁶ m (FIG. 25), 2·10⁻⁵ m (FIG.26). The heating power as shown in FIGS. 24, 25 and 26 depends on afrequency of the AC power. As frequency increases, the amount of appliedpower used to melt an interfacial layer of ice levels off. The ACvoltage was about 500 v. At a coating thickness of about 10 μm (10⁻⁵ m),the respective heating power for water and ice are substantially equal,as is shown from FIG. 24.

[0255] FIGS. 27-29 graphically illustrate certain test analyses ofcircuit 200 in which circuit 200 is applied to a slider, such as thosedescribed in more detail below. In this embodiment, a change in snowtemperature under the slider is taken into consideration. Circuit 200may be characterized by the following Table 26-1 (MathCad file): TABLE27-1 $\rho:={300\frac{kg}{m^{3}}}$

x := 0, 0.0001 . . . 0.1 m C := 2 · 10³ J/kg K $\begin{matrix}{\lambda:={0.2\frac{2}{\overset{.}{m}k}}} \\{W:={{1 \cdot 10^{3}}\frac{w}{m^{2}}}}\end{matrix}\quad$

$D:=\frac{\lambda}{C \cdot \rho}$

D = 3.333 × 10⁻⁷${y\left( {x,t} \right)}:=\frac{x}{\sqrt{4 \cdot D \cdot t}}$

${\Delta \left( {x,t} \right)}:={\frac{W}{\lambda} \cdot \sqrt{4 \cdot D \cdot t} \cdot {\int_{y{({x,t})}}^{\infty}{\left( {1 - {{erf}(z)}} \right){z}}}}$

t := 0, 0.001 . . . 1 s a := 0.1 m L := 1.5 m v := 1, 2 . . . 30${W_{speed}\left( {\Delta,v} \right)}:={a \cdot \Delta \cdot \sqrt{v \cdot \lambda \cdot C \cdot L \cdot \rho}}$

W_(speed)(1, 10) = 134.164 watt,

[0256] where

[0257] ρ is the snow density, x is the distance inside the snow from theslider, C is the heat capacitance of the snow, λ is a thermalconductivity coefficient of the snow, W is the heating power, D is thethermal diffusivity of snow, t is a duration in which power is applied,a is a slider width, L is a length of the slider, V is a slidervelocity, y is an integration variable, W_(speed) is the heating powerwith respect to speed of the slider, and Δ is overheating temperature Δ.

[0258]FIG. 27 illustrates dependence of snow overheating temperature Δ(e.g., degrees C.) with respect to the distance from a slider. In FIG.27, Y-axis 295 represents overheating temperature Δ (° C.) and X-axis294 represents distance from the slider (in meters). With a heatingpower W of about 1 kwatt/m², plots 290, 291, 292, and 293 illustratetemperature dependences for heating pulses having approximate durationsof t=0.1 s, 0.2 s, 0.5 s, and 1 s, respectively. FIG. 28 illustrates thesnow-slider interface-temperature dependency with respect to time (plot300) when HF-power of density 1000 watt/m² was applied. In FIG. 28,Y-axis 301 represents overheating temperature Δ (° C.) and X-axis 302represents time (in seconds).

[0259]FIG. 29 illustrates the heating power required to increase theinterface temperature by 1° C. when the slider is traveling at velocityv of about 30 m/s. In FIG. 29, Y-axis 311 represents heating powerW_(speed) and X-axis 312 represents velocity ν. In the example, as theslider travels at about 5 m/s, the heating power is about 100watts. Theheating power W_(speed) is plotted with respect to velocity ν (plot310).

[0260] FIGS. 30-35 show graphs illustrating one analysis of heattransfer through convection of one de-icer system and heat transferthrough a substrate of one HF de-icer system. In the example, astationary solution (e.g., constant power) is exemplarily characterized.FIG. 30 shows a dependence of a heat transfer coefficient h_(c) on airvelocity (plot 320) assuming a cylindrical aerofoil (the leading edge ofan aircraft wing). In FIG. 30, Y-axis 321 represents heat transfercoefficient h_(c) and X-axis 322 represents velocity ν. The heattransfer coefficient h_(c) for the aerofoil may be calculated accordingto the following Table 30-1: TABLE 30-1 (MathCad file) v := 89 D := 0.03v := 10, 11 . . . 100${h_{c}\left( {v,D} \right)}:={{\frac{v^{0.63}}{D^{0.37}} \cdot \frac{0.190 \cdot 0.024 \cdot 0.69^{0.36}}{\left( {1.2 \cdot 10^{- 5}} \right)^{0.63}}}\quad {{watt}/m^{2}}K}$

h_(c)(89, 0.254) = 141.057 watt/m² K h_(c)(89, 0.0254) = 330.669 watt/m²K,

[0261] where

[0262] v is air velocity and D is an aerofoil diameter. Approximatelyhalf of the heat transfer coefficient h_(c) may be attributed to a frontsection of the aerofoil when using a Reynolds number of about 1.9×10⁵.

[0263] In one example, a heat transfer coefficient h_(c) of about 165watt/m² K used in an HF de-icer generates a power W of about 4.5 kwattsper square meter. The de-icer includes a polymer layer of thickness dwith a thermal conductivity coefficient of λ_(d). Ice is grown on thede-icer with a thickness L. The ice thermal conductivity coefficient isλ and the thickness of the heated interfacial layer of ice is about oneinter electrode spacing, or about 0.25 mm. A steady-state overheatingtemperature of the interfacial layer of ice Δ=T_(i)−T_(a), where T_(i)is the interface temperature and T_(a) is the ambient temperature, maybe calculated according to the following Table 30-2 (Math Cad file):TABLE 30-2 $W:={4500\frac{w}{m^{2}}}$

d := 0.002 m $h:={165\quad \frac{w}{m^{2}K}}$

L := 0, 0.0001 . . . 0.01 m l := 0.00001, 0.00002 . . . 0.001 m$\begin{matrix}{\lambda:={2.22\frac{m}{mK}}} \\{\lambda_{d}:={0.35\frac{w}{mK}}}\end{matrix}\quad$

${\Delta \left( {L,l,\lambda_{d}} \right)}:={W \cdot d \cdot \left\lbrack \frac{{h \cdot L} + {\left( {\lambda - {l \cdot h}} \right) \cdot \left( {1 - {\exp \left( \frac{- L}{l} \right)}} \right)}}{{h \cdot \left( {{\lambda \cdot d} + {L \cdot \lambda_{d}}} \right)} + {\lambda \cdot \lambda_{d}}} \right\rbrack}$

[0264]FIG. 31 shows a dependence of the steady-state (stationarysolution) overheating Δ in ° C. on ice thickness in meters. In FIG. 31,Y-axis 335 represents overheating Δ and X-axis 336 represents thicknessL. Plot 330 shows a dependence of steady-state overheating in ° C. onice thickness in meters assuming a theoretically perfect insulatinglayer between the de-icer and the aerofoil, while plot 331 shows thedependence for a 2 mm thick Teflon film between the de-icer and theaerofoil. De-icing performance is maximized when ice thickness exceedsapproximately 1 mm (point 333 for the theoretically perfect insulatinglayer, and point 334 for the 2 mm thick Teflon film).

[0265]FIG. 32 shows a dependence of the steady-state overheating Δ in °C. on electrode size in meters (plot 340), assuming a perfect insulatinglayer and a 1 cm thickness of ice. In FIG. 32, Y-axis 341 representsoverheating Δ and X-axis 342 represents electrode size l. In theexample, bubbling on the interfacial layer of ice may be seen. Bubblingis the result of ice evaporation (e.g., steam) and is evidence ofoverheating by more than 110° C.

[0266] When used in operational environments, the de-icer may have aperformance that is better than performances achieved in laboratoryenvironments. For example, atmospheric ice growing on an aerofoil hasphysical properties that differ from those of solid ice. Atmospheric icecan include unfrozen water and/or gas bubbles. These additions toatmospheric ice may reduce ice thermal conductivity and density. Toillustrate, the thermal conductivity of water is approximately 0.56 w/mKas opposed to the thermal conductivity of solid ice at approximately2.22 w/mK. An interfacial layer of ice (e.g., a layer of ice adjacent tothe de-icer) is warmer than remaining ice and may contain water.

[0267] A heat exchange de-icer used in operational environmentalconditions may be modeled by approximating ice thermal conductivitycoefficient λ as a number between about 0.5 w/mK and 2.22 w/mK. Anexample is calculated according to the following Table 30-3: TABLE 30-3$W:={4500\frac{w}{m^{2}}}$

d := 0.002 m $h:={165\quad \frac{w}{m^{2}K}}$

L := 0, 0.0001 . . . 0.01 m l := 0.00001, 0.00002 . . . 0.001 m$\begin{matrix}{\lambda:={1\frac{m}{mK}}} \\{\lambda_{d}:={0.35\frac{w}{mK}}}\end{matrix}\quad$

${\Delta \left( {L,l,\lambda_{d}} \right)}:={W \cdot d \cdot \left\lbrack \frac{{h \cdot L} + {\left( {\lambda - {l \cdot h}} \right) \cdot \left( {1 - {\exp \left( \frac{- L}{l} \right)}} \right)}}{{h \cdot \left( {{\lambda \cdot d} + {L \cdot \lambda_{d}}} \right)} + {\lambda \cdot \lambda_{d}}} \right\rbrack}$

[0268]FIG. 33 shows a dependence of the steady-state (stationarysolution) overheating Δ in ° C. on ice thickness in meters. In FIG. 33,Y-axis 355 represents overheating Δ and X-axis 356 represents thicknessL. Plot 350 shows a dependence of steady-state overheating in ° C. onice thickness in meters assuming a theoretically perfect insulatinglayer between the de-icer and the aerofoil, while plot 351 shows thedependence for a 2 mm thick Teflon film between the de-icer and theaerofoil. De-icing performance is maximized when ice thickness exceedsapproximately 1 mm (point 352 for the theoretically perfect insulatinglayer and point 353 for the 2 mm thick Teflon film).

[0269] Inhomogeneous electric power distribution near de-icingelectrodes may also cause bubbling of the interfacial layer of ice. Forexample, an electrode's surface local density of power can exceed themean power by about one order of magnitude due to variations in electricfield strength. As such, at locations of where power exceeds the meanpower, the electrode may heat the interfacial layer of ice more rapidlythan at other locations to generate steam.

[0270] Results of a time dependent solution may vary from those of thesteady-state solutions. For example, since ice is a material with lowthermal diffusivity coefficient, as HF power is applied to aninterfacial layer of ice, a “heat wave” propagates through the ice.Accordingly, a thin layer of ice may be considered to be a thermallyisolated layer of ice. As such, the de-icer may predominantly applypower to only that layer. Time dependent temperature curves Δ(x,t)(plots 360, 361, 362 and 363 of FIG. 34) may be calculated according tothe following Table 30-4: TABLE 30-4 (MathCad file)$\rho:={920\frac{kg}{m^{3}}}$

$C:={{2 \cdot 10^{3}}\quad \frac{J}{{kg} \cdot K}}$

x := 0, 0.0001 . . . 0.1 m $\lambda:={1\frac{w}{mK}}$

$\begin{matrix}{W:={{4.5 \cdot 10^{3}}\frac{w}{m^{2}}}} \\{D:=\frac{\lambda}{\rho {\cdot C}}} \\{{y\left( {x,t} \right)}:=\frac{x}{\sqrt{4 \cdot D \cdot t}}}\end{matrix}\quad$

${\Delta \left( {x,t} \right)}:={\frac{W}{\lambda} \cdot \sqrt{4 \cdot D \cdot t} \cdot {\int_{y{({x,t})}}^{\infty}{\left( {1 - {{erf}(z)}} \right){z}}}}$

t := 0, 0.1 . . . 1000 s D = 5.435 × 10⁻⁷,

[0271] where

[0272] ρ is ice density, C is ice heat capacity of the ice, λ is athermal conductivity coefficient of the ice, x is a distance from theheater, W is an applied power per square meter, D is a coefficient ofheat diffusivity, and t is the duration in which power is applied (e.g.,as a heat pulse). FIG. 34 illustrates plots 360, 361, 362 and 363 forrespective time values of 200 s, 100 s, 25 s and 5 s as the power W ofabout 4.5 kwatt/m² is applied to an atmospheric ice mixture of solidice, unfrozen water and gas bubbles with a thermal conductivitycoefficient λ of 1 W/m □K. In FIG. 34, Y-axis 365 represents overheatingΔ and X-axis 366 represents distance from the heater x.

[0273] Interface temperature (i.e., the temperature of an interfaciallayer of ice) has a typical diffusion time T as calculated according tothe following Table 30-5: TABLE 30-5 (MathCad file) L := 10⁻²$\tau:=\frac{L^{2}}{D}$

τ = 184 s

[0274]FIG. 35 illustrates how the interface temperature depends on timeby showing a dependence of interfacial overheating temperature Δ in ° C.on time. In FIG. 35, Y-axis 371 represents overheating Δ and X-axis 372represents time. When a short pulse of heating is applied, thermalenergy can be minimized and still melt the interfacial layer of ice. Forexample, thermal energy may be calculated according to the followingTable 30-6: TABLE 30-6(MathCad file)${\Delta \left( {x,t} \right)}:={\frac{W}{\lambda} \cdot \sqrt{4 \cdot D \cdot t} \cdot {\int_{0}^{\infty}{\left( {1 - {{erf}(z)}} \right){z}}}}$

${\Delta (t)}:={2 \cdot \frac{W}{\lambda} \cdot \sqrt{\frac{D \cdot t}{\pi}}}$

${t(\Delta)}:={\left( \frac{\Delta \cdot \lambda}{2 \cdot W} \right)^{2} \cdot \frac{\pi}{D}}$

${Q(W)}:={\left( \frac{\Delta \cdot \lambda}{2} \right)^{2} \cdot \frac{\pi}{D \cdot W}}$

[0275] where t is the time it takes to reach a desired overheatingtemperature Δ of the interfacial layer of ice, and Q is the totalthermal energy needed to reach that temperature. As in FIG. 1, totalthermal energy Q may be substantially inversely proportional to appliedpower W, to employ a de-icer with a higher power output that conservesoverall electric power.

Thermal Transfer De-Icer Systems

[0276] In the following embodiments, thermal transfer de-icer systemsare described. The thermal transfer de-icer systems may be used toremove ice from a surface of an object. In some embodiments, thefollowing systems may also be used to melt an interfacial layer of iceand modify a coefficient of friction of an object's surface to iceinterface. In one example, such thermal transfer de-icer systems storethermal energy and intermittently transfer the thermal energy from aheating source (or heating supply) to a heating element.

[0277]FIG. 36 shows one thermal transfer de-icer system 460, in accordwith one embodiment. Thermal transfer de-icer system 460 is illustratedin two states, 460A and 460B. Thermal transfer de-icer system 460includes power supply 464, thermal insulator 462, heating element 466,membrane 470, and membrane valve 468. Thermal transfer de-icer system460 is configured for removing ice 472 from a surface (e.g., includingouter surface 471 of membrane 470 ) of an object such as an aircraft, anaircraft wing, a tire, an automobile windshield, a boat, an aircraft, aroad, a bridge, a sidewalk, a freezer, a refrigerator, building, arunway, and a window. Thermal transfer de-icer system 460 may provideheat storage such that once the heat is stored it is applied as heatpulses to the ice-object interface, as desired. Power supply 464 mayinclude a switching power supply, battery, a capacitor, a flywheel,and/or a high-voltage power supply. The capacitor may be a supercapacitor or an ultracapacitor.

[0278] In state 460A, membrane 470 is inflated with gas through membranevalve 468. A typical gas may include air or other gases with thermalinsulating properties. The application of the power to heating element466 converts the power into a magnitude of thermal energy that is storedin heating element 466. Thermal energy stored in heating element 466 istransferred to interfacial layer 473 by deflating membrane 470, as shownin state 460B. When membrane 470 is deflated, the thermal energy istransferred from heating element 466 to interfacial layer 473 to meltinterfacial layer 473, so that ice 472 is removed. In one embodiment,state 460B is maintained just long enough to melt the interfacial layerof ice 472

[0279]FIG. 37 shows one thermal transfer de-icer system 480 in accordwith one embodiment. Thermal transfer de-icer system 480 is illustratedin two states, 480A and 480B. Thermal transfer de-icer system 480includes power supply 484, thermal insulator 486, and heating element482. Thermal transfer de-icer system 480 is configured for removing ice492 from surface 491 of an object 493. Object 493 may be of the class ofobjects discussed herein. Thermal transfer de-icer system 480 mayprovide heat storage such that once the heat is stored it can be appliedas heat pulses to the ice-object interface at surface 491, as desired,to melt interfacial ice.

[0280] In state 480A, heating element 482 is shown as two layers, 482Aand 482B, that “sandwich” thermal insulator 486. Thermal insulator 486is moveably attached between heating element layers 482A and 482B suchthat both layers slide into contact with one another as shown in state480B. Power supply 484 applies a magnitude of power to heating element482. Power supply 484 may be one or more of power supplies described inFIG. 36. The application of the power to heating element 482 convertsthe power into thermal energy. When layer 482A is in contact with layer482B, the thermal energy transfers from heating element 482 to aninterfacial layer of ice 492 in an amount sufficient to melt thatinterfacial layer. In one embodiment, heating element layers 482A and482B are frequently moved across one another such that thermal insulator486 periodically thermally isolates layers 482A and 482B and causes aperiodic transfer of thermal energy to the interfacial layer of ice atsurface 491. The periodic transfer of thermal energy provides an averageenergy to the interfacial layer to keep the object free of ice.

[0281] Heating element 482 may be formed of a conductive material suchas metal, a metal alloy foil, a thin metal layer on a dielectricsubstrate, a thin metal oxide layer on a substrate, a conductive polymerfilm, a conductive paint, a conductive adhesive, a wire mesh andconductive fibers. Examples of transparent conductors include SnO2, ITO,TiN, and ZnO. Examples of conductive fibers include carbon fibers.

[0282]FIG. 38 shows one thermal transfer de-icer system 500 in accordwith one embodiment. Thermal energy transfer de-icer 500 includes powersupply 504, heating element 502, water pump 508, tank 506, and tube 510.Thermal transfer de-icer system 500 is configured for removing ice 512from a surface 511 of an object. Thermal transfer de-icer system 500 mayoperate as a heat storage such that once the heat is stored it can beapplied as a heat pulse to the ice-object interface at surface 511.

[0283] Power supply 504 applies power to heating element 502. Powersupply 504 may be one or more of power supplies described in FIG. 36.The application of the power to heating element 502 converts the powerinto thermal energy. Heating element 502 raises a temperature of athermally conductive liquid in tank 506. The thermally conductive liquidmay include water or some other thermally conductive liquid. Thethermally conductive liquid is pumped through tube 510 with pump 508.The thermal energy is transferred to an interfacial layer of ice 512 atsurface 511 as the thermally conductive liquid is pumped into tube 510.As the thermal energy is transferred to the interfacial layer, theadhesion of ice 512 is disrupted from surface 511. In one embodiment,the thermally conductive liquid is frequently pumped through tube 510with pump 508 to cause a substantially periodic transfer of thermalenergy to the interfacial layer, to provide and average thermal energyto the interface to keep the object free of ice.

[0284]FIG. 39 shows pulse de-icer system 520; system 520 is shown tocontrast differences between thermal transfer de-icer systems of inFIGS. 37 and 38 with previously described systems (e.g., system 10 ofFIG. 1). In this embodiment, ice 528 illustratively adheres to a surface531 at the object-ice interface adjacent surface 531. Pulse de-icersystem 520 includes power supply 524, one or more heating elements 526,and layers 522A and 522B. Pulse de-icer system 520 is configured forremoving ice 528 from a surface 531 of layer 522B. For example, layer522B is an object, such as windshield, to be de-iced.

[0285] Heating elements 526 are embedded in layer 522B and electricallyconnected to power supply 524, to receive power therefrom. In oneexample, layers 522A and 522B are formed of a substantially transparentmaterial for use in or as a windshield. As power supply 524 appliespower to heating elements 526 (which may also be transparent), thermalenergy radiates from heating elements 526 and disrupts an adhesion ofice 528 to surface 531 of layer 522B. In one embodiment, power supply524 applies power to heating elements 526 according to the equations ofFIG. 1. Power supply 524 may be one or more of power supplies describedin FIG. 36, for example.

[0286] The application of power to heating elements 526 thus convertsthe power into a magnitude of thermal energy. The thermal energy istransferred to an interfacial layer of ice 528 at surface 531 to disruptthe adhesion of ice 528 onto surface 531. In one embodiment, the poweris frequently pulsed to heating elements 526 to cause a substantiallyperiodic transfer of thermal energy to the interfacial layer for aperiodic duration such as described in Eq. 1-1.

[0287] In comparison, a power supply of a thermal transfer de-icersystem (e.g., power supplies 484 and 504 of FIGS. 37 and 38,respectively) delivers power to heating elements which in turn producethermal energy. The thermal transfer de-icer sytsem then stores thethermal energy until applied as thermal energy to the ice-to-objectinterface.

[0288] Heating elements 526 of pulse de-icer system 520 may be made of ametal, metal alloy foil, a thin metal layer on a dielectric substrate, athin metal oxide layer on a substrate, a substantially transparentconductor, a conductive polymer film, a conductive paint, a conductiveadhesive, a wire mesh and/or conductive fibers, for example. Examples oftransparent conductors include SnO2, ITO, TiN, and ZnO. Examples ofconductive fibers include carbon fibers. Heating elements 526 may alsoinclude semiconductor devices configured for converting the power intothermal energy. By using multiple heating elements 526, total energyrequirements can be segmented or discretely determined. For example, asegment 535 of surface 531 requires substantially less energy to melt aninterfacial layer of ice in that region as compared to melting aninterfacial layer of ice for all of surface 531. Accordingly,instantaneous power requirements for disrupting the adhesion of ice 528are decreased as sequential pulsing across the segments or sectionsdiscretely disrupts ice 528 from all of surface 531, over time.

[0289]FIG. 40 shows one thermal transfer de-icer system 540 in accordwith one embodiment. Thermal transfer de-icer system 540 includesthermal conductor 542 (e.g., a “hot plate”), dielectric plate 546, andheated element 544 (e.g., thin metal foil). Thermal transfer de-icersystem 540 is configured for melting an interfacial layer of ice 545 onan object by pulsating thermal energy to ice 545. For example, thermaltransfer de-icer system 540 may be positioned with a surface of anobject such that when heating power is applied to heated element 544, aninterfacial layer of the ice 545 is melted.

[0290] In one embodiment, thermal conductor 542 converts power intothermal energy. The thermal energy is transferred from thermal conductor542 to heating element 544 through holes 547 in dielectric plate 546. Inone example, thermal conductor 542 vibrates such that when thermalconductor 542 contacts heating element 544, thermal conductor 542transfers thermal energy to heating element 544, which in turn melts aninterfacial layer of ice. Depending on the application of thermaltransfer de-icer system 540, melting the interfacial layer of the icemay be useful to remove ice from a surface of an object, to prevent itsformation on the surface, or to modify its adhesion strength and changea coefficient of friction between the ice and the object.

[0291] In one embodiment, thermal transfer de-icer system 540 is used asa “pulse brake” in which a heating pulse is transferred from thermalconductor 542 to heating element 544 when thermal conductor 542 touchesheating element 544 affixed to a base of a slider, which interfaces theice. When braking is needed, thermal conductor 542 touches the heatingelement 544 for few milliseconds, through holes 547 in dielectric plate546, creating “hot spots” where ice melts. After thermal conductor 542is withdrawn, the melted spots typically freeze within a fewmilliseconds, providing bonds between the slider base and the ice.

[0292] One parameter of a pulse brake is the time it takes for ice/snowto melt and then refreeze. When interfacial cooling occurs between iceor snow and the slider base, that time, t_(cool), may be estimated as:$\begin{matrix}{{t_{cool} \approx \left\lbrack {\frac{Q}{S} \cdot \frac{1}{\left( {T_{m} - T} \right)\left( {\sqrt{\lambda_{snow} \cdot \rho_{snow} \cdot c_{snow}} + \sqrt{\lambda_{ski} \cdot \rho_{ski} \cdot c_{ski}}} \right)}} \right\rbrack^{2}},} & \text{(Eq.~~40-1)}\end{matrix}$

[0293] where

[0294] T_(m) is an ice melting temperature, T is an ambient temperature,λ is a thermal conductivity coefficient, ρ is the material density, andc is the material heat capacity (subscript “snow” denotes ice and/orsnow and subscript “ski” denotes a material used as the slider base) Wis a power per square meter, Q is the thermal energy to be dissipated,and S is the slider base area.

[0295]FIG. 41 shows one thermal transfer system 560, which was built andtested, in accord with one embodiment of FIG. 36. In this embodiment,thermal transfer system 560 includes two aluminum discs 562 and 563 ofabout six inches in diameter and 1 mm thick. In one embodiment, interiorsurfaces of discs 562 and 563 are lapped and buffed to decrease opticalemittance. Exterior surfaces of discs 562 and 563 are anodized in abouta 15% sulfuric acid solution to achieve a thickness of about 10 μm to 12μm of aluminum oxide film (hard anodizing). Discs 562 and 563 areattached to Plexiglas ring 569 by a rubber O-ring 570B. Discs 562 and563 are further attached to Plexiglas ring 572, and thus valve 571, byrubber O-ring 570A.

[0296] Thermal transfer system 560 also includes heating element 565affixed to disc 563 and configured to receive electrical power frompower supply 566, to convert that power into thermal energy. Heatingelement 565 includes a carbon foil encapsulated into Kapton polyimidsubstrate 568. Thermocouple 564 may be affixed to disc 563 through ahole 579 in heating element 565 by means of thermoconductive glue. Inthis embodiment, thermocouple 564 is configured to control thetemperature of disc 562 as heating element 565 transfers heat to disc563. In one embodiment, power supply 566 is a DC power supply configuredto supply about 20V.

[0297] A vacuum pump may physically couple to valve 571 to bring “cold”and “hot” discs into contact and to transfer thermal energy from a hotdisc to a cold disc. For example, as power supply 566 supplies power toheating element 565, heating element 565 converts the power into thermalenergy and transfers that energy to disc 563, thereby creating a hotdisc. The vacuum pump withdraws air from chamber 573 to collapse chamber573 and to bring disc 562 into contact with disc 563 (e.g., the colddisc). Once disc 562 contacts disc 563, thermal energy of disc 563transfers to disc 562. When the transfer of thermal energy is no longerdesired, the vacuum pump inflates chamber 573 with air to separate thediscs 562 and 563.

[0298] At about −10° C., and with ice grown on disc 562 and thermaltransfer system 560 in a vertical position, a power of approximately10-25 watts heats disc 563 to about 20° C. when applied to heatingelement 565. When the vacuum pump withdraws air from chamber 573, suchthat discs 562 and 563 contact one another, ice 577 is removed from disc562, e.g., by gravity. While air is typically used in chamber 573, otherthermally insulating gases may alternatively be used in chamber 573.

Thermal Transfer De-Icer System Analysis

[0299] In the following description, various thermal transfer de-icersystems are analyzed and their operative characteristics shown. Forexample, characteristics of various materials are analyzed, such as iceat certain temperature having a known capacitance (e.g., C_(i) of FIG.18). In these analyses, the component values illustrate variousconditions, such as environmental conditions and/or heat transfermethods.

[0300] FIGS. 42-46 show graphs illustrating one exemplary analysis of athermal transfer de-icer system. In the example, a thermal transferde-icer system has a first and second thermal conductor and a heatingelement with equal heat capacities. The system is characterized with anatural convection heat exchange Nu across an air gap in which theheating element heats the first thermal conductor to cause the secondthermal conductor to reach a temperature of about 275.5K when the twothermal conductors contact one another. Such a system can becharacterized by the following Table 42-1 (calculating Nusselt numberfor natural convection of air between the discs 562, 563 of FIG. 41):TABLE 42-1 (MathCad file) ν := 1.57 · 10⁻⁵ L := 0.0125 g := 9.8$\beta:=\frac{1}{273}$

Pr := 0.69 T_(m) := 273 T_(s) := 243, 244 . . . 273 T_(h)(T_(s)) := 2 ·T_(m) − T_(s) + 5 Δ(T_(s)) := T_(h)(T_(s)) −T_(s) (i.e., the temperaturedifference between the heater and the environment)${{Ra}_{L}\left( T_{s} \right)}:=\frac{g \cdot \beta \cdot L^{3} \cdot \Pr \cdot \left( {\Delta \left( T_{s} \right)} \right)}{v^{2}}$

Ra_(L)(243) = 1.276 × 10⁴${{Nu}_{1}\left( T_{s} \right)}:={0.0605\quad {{Ra}_{L}\left( T_{s} \right)}^{\frac{1}{3}}}$

${{Nu}_{2}\left( T_{s} \right)}:=\left\lbrack {1 + \left\lbrack \frac{0.014\quad {{Ra}_{L}\left( T_{s} \right)}^{0.293}}{1 + \left( \frac{6310}{{Ra}_{L}\left( T_{s} \right)} \right)^{1.36}} \right\rbrack^{3}} \right\rbrack^{\frac{1}{3}}$

[0301] where T_(s) is the temperature of the substrate material (disc562), T_(h) is the temperature of the heating element (disc 563), ν isair kinematic viscosity, L is a distance between discs 562 and 563, g isgravity acceleration, β is air thermal expansion coefficient, Pr is airPrandtl number, T_(m) is ice melting temperature, T_(s) is anincremental temperature of disc 562, Δ is temperature difference, Ra isair Rayleigh number, Nu₁ and Nu₂ are Nusselt number.

[0302] Accordingly, FIG. 42 shows (in plot 580) a dependence of Nusseltnumber on outside temperature (cold disc 562). Table 42-2 calculatesnatural convection heat transfer rate between the discs 562, 563.: TABLE42-2 (MathCad file) λ_(a) := 0.025${W_{c}\left( T_{s} \right)}:=\frac{\lambda_{a} \cdot {{Nu}\left( T_{s} \right)} \cdot {\Delta \left( T_{s} \right)}}{L}$

$\frac{W_{c}(243)}{2} = {91.887\quad \frac{watt}{m^{2}}}$

[0303] where λ_(a) is a thermal conductivity coefficient of the air, andW_(c)/2 is a mean heat transfer rate when the heater heats disc 563 fromT_(s) to T_(h). In FIG. 42, Y-axis 581 represents convection Nu andX-axis 582 represents temperature T_(s) of the substrate material. Amean loss of heat W_(c) through the air gap is shown in FIG. 43 (plot590). In FIG. 43, Y-axis 591 represents convection heat transfer W_(c)/2and X-axis 592 represents temperature T_(s) of the substrate material.

[0304]FIG. 44 illustrates heat transfer Win, through back insulation(e.g., insulation backing the first thermal conductor, plot 600). Inthis embodiment, the insulation is a rigid polyurethane foam with athickness l of about 0.025 m and a thermal conductivity coefficientλ_(a) of about 0.026. The heating transfer W_(in) can be calculatedaccording to the following Table 42-3 (heat loss through the backinsulating layer): TABLE 42-3 (MathCad File)${W_{in}\left( T_{s} \right)}:=\frac{\lambda_{in} \cdot {\Delta \left( T_{s} \right)}}{1}$

$\frac{W_{in}(243)}{2} = {33.8\quad \frac{watt}{m^{2}}}$

[0305] Accordingly, radiative heat transfer W_(R) through the air gapmay be calculated according to the following Table 42-4 (heat lossthrough radiation): TABLE 42-4 (MathCad File) ε := 0.04 σ := 5.67 · 10⁻⁸W_(R)(T_(s)) := ɛ ⋅ σ ⋅ (T_(h)(T_(s))⁴ − T_(s)⁴)

${W_{R}(243)} = {12.502\quad \frac{watt}{m^{2}}}$

[0306] where ε is the emittance of discs 562 and 563 emittance, and σ isthe Stefan-Boltzmann constant. From Table 42-4, the radiative heattransfer W_(R) can be plotted (plot 600) as a function of temperatureT_(s) in FIG. 44 (T_(s) and T_(m) being defined above). In FIG. 44,Y-axis 601 represents radiative heat transfer W_(R) and X-axis 602represents temperature T_(s) of the substrate material.

[0307]FIG. 45 illustrates a total mean heat loss W (plot 610) from theheating element. In FIG. 45, Y-axis 611 represents total mean heat lossW and X-axis 612 represents temperature T_(s) of the substrate material.Because temperature of the heating element cycles between T_(m) andT_(h), a mean difference in the temperature between the heating elementand the environment is approximately (¾)*(T_(h)−T_(s)). The total meanheat loss W may be calculated according to the following Table 42-5(total heat loss to the environment): TABLE 42-5 (MathCad file)${W\left( T_{s} \right)}:={\frac{3}{4} \cdot \left( \left( {{W_{c}\left( T_{s} \right)} + {W_{in}\left( T_{s} \right)} + {W_{R}\left( T_{s} \right)}} \right) \right)}$

${W(243)} = {197.907\quad \frac{watt}{m^{2}}}$

${W(253)} = {127.163\quad \frac{watt}{m^{2}}}$

${W(263)} = {63.602\quad \frac{watt}{m^{2}}}$

[0308]FIG. 46 illustrates a mean power W_(m) from a power supply used inone thermal transfer de-icer system. In FIG. 46, Y-axis 623 representsmean power W_(m) and X-axis 624 represents temperature. The mean powerresults are shown as a function of three ambient cold plate temperaturesT_(s) (plots 620, 621 and 622). The total amount of thermal energy Qthat it takes to heat the heating element from the temperature T_(s) ofthe substrate material to T_(h) is calculated as two components, Q1 andQ2. Q1 is thermal energy due to heat capacity of the heating element andQ2 is thermal energy that is transferred from the heater to theenvironment (total energy loss from the system). The total amount ofthermal energy Q may be calculated according to the following Table42-6: TABLE 42-6 (MathCad file) d := 0.001 t := 1, 2 . . . 300 C_(s) :=900 λ_(s) := 170 ρ_(s) := 2700 C_(i) := 2000 ρ_(i) := 920 λ_(i) := 2Q₁(T_(s)) := d · C_(s) · ρ_(s) · (T_(h)(T_(s)) − T_(m))${Q_{1}(243)} = {8.505 \times 10^{4}\quad \frac{Joul}{m^{2}}}$

Q₂(T_(s), t) := W(T_(s)) · t

[0309] where

[0310] d is the heating element thickness, t is the duration in whichheat is exchanged (e.g., for a heat pulse), C is the material heatcapacity, λ is a thermal conductivity coefficient, ρ is the materialdensity (subscript “i” denotes ice and/or snow and subscript “s” denotessubstrate material for most aluminum alloys), T_(s) is the temperatureof the substrate, T_(h) is the temperature of the heating element, andT_(m) is the ice temperature. The mean power from a power supply used inthis example thermal transfer de-icer system (de-icing every threeminutes (180 s)) may be calculated according to the following Table42-7: TABLE 42-7 (MathCad file)${W_{m}\left( {t,T_{s}} \right)}:={\frac{Q_{1}\left( T_{s} \right)}{t} + {W\left( T_{s} \right)}}$

${W_{m}\left( {180,243} \right)} = {670.407\quad \frac{watt}{m^{2}}\quad \left( {{e.g.},{{plot}\quad 620}} \right)}$

${W_{m}\left( {180,253} \right)} = {464.663\quad \frac{watt}{m^{2}}\quad \left( {{e.g.},{{plot}\quad 621}} \right)}$

${W_{m}\left( {180,263} \right)} = {266.102\quad \frac{watt}{m^{2}}\quad \left( {{e.g.},{{plot}\quad 622}} \right)}$

[0311] In one example, the de-icer system with the above characteristicsis useful with an aerofoil (e.g., aircraft wing) de-icer. Such a de-icersystem can be made of 1 mm thick aluminum alloy and attached behind aleading edge of a small aerofoil (i.e., the forward exposed portion ofan aircraft wing). In the example, the aerofoil has a span of about 20cm and thickness of about 5 cm; the de-icer dimensions are about 20cm×10 cm. At an air speed of about 142 km/h, and at approximately −10°C. with about 20 cm water droplets, atmospheric ice forms on theaerofoil. After ice growth of about 5 mm to 10 mm thickness, a computersystem (e.g., controller 78 of FIG. 6) directs a power supply to applypower to the de-icer to melt an interfacial layer of ice on the aerofoilsurface, such that the adhesion of the ice to the aerofoil surface issubstantially modified and/or broken. The ice can then be removed fromthe aerofoil surface by air drag force. Such an example aerofoil systemwas built and tested, demonstrating a performance which was very closeto theoretical predictions of Table 42-7.

Methods of Coefficient of Friction Manipulation

[0312] In the following embodiments, the coefficient of friction ismodified between an object surface (e.g., as part of a slider) and iceor snow. In one example, a system such as system 40, FIG. 4, employs theequations of FIG. 1 to affect the coefficient of friction between aslider and snow (e.g., as described in connection with FIGS. 47 and 48).Such a system can assist in increasing or decreasing traction betweenthe surface interface and the snow as determined by a particularapplication. For example, certain sliders described herein employ such asystem as a pulse brake to brake the slider as it travels across snow.

[0313]FIGS. 47 and 48 illustrate characteristics of a slider, such as aski or an automobile tire, in accord with one embodiment, The sliderincludes slider substrate 632 and heating element 633. Heating element633 is affixed to slider substrate 632 and may be in direct contact withice and/or snow 630. Heating element 633 is configured for receivingpower from a power source, for example in accordance with the equationsof FIG. 1.

[0314]FIG. 48 illustrates temperature diffusion within slider substrate632 and ice 630 when power is applied to heating element 633 in the formof a pulse. For example, FIG. 48 illustratively shows a heat diffusiondistance along X-axis 636 over a given time t₁ through ice 630 andsubstrate 632, as a function of temperature change T along T-axis 639 atthe ice-object interface. The curve t₁ represents temperature changecaused by heat diffusion into ice 630 and into substrate 632 for a givenpulse duration. As shown, the peak of curve t₁ is at a certaintemperature 638 on T-axis 639; temperature 638 is sufficient to melt aninterfacial layer of ice 630. The shaded area (m) under the curve t₁represents a melted interfacial layer.

[0315] Prior to applying power to heating element 633, the ambienttemperature is represented by point 637. Once a pulse of power isapplied to heating element 633, temperature of element 633 begins torise, and transfers into ice 630 for a distance 631 (the distance of aninterfacial layer of ice 630) and into substrate 632. This temperaturerises to point 635 where ice begins to melt and continues to rise forthe duration in which pulse power is applied. Thermal energy melts athin interfacial layer (m) of ice 630. Once the power is removed fromheating element 633, the temperature begins to drop below the meltingpoint 635, curve t_(2.) As the temperature of heating element 633decreases, the adhesion of ice 630 to slider substrate 632 is modifieddue to refreezing. This refreezing increases the adhesion of the ice 630to substrate 632 and assists in braking the slider at the interface ofheating element 633.

[0316] In one embodiment, the characteristics of the slider conform tothe equations of FIG. 10. For example, the diffusion time t over alength L coincident with X-axis 636 may be in the form: $\begin{matrix}{{t = \frac{L^{2}}{D}},} & \text{(Eq.~~11-1)}\end{matrix}$

[0317] where

[0318] D is a coefficient of heat diffusivity set forth by:$\begin{matrix}{{D = \frac{\lambda}{\rho \quad c}},} & \text{(Eq.~~11-2)}\end{matrix}$

[0319] where

[0320] λ is a thermal conductivity coefficient, ρ is the materialdensity, and c is the material heat capacity. Accordingly, equations.11-1 and 11-2 illustrate that heat energy captured inside ice 630 andsubstrate 632 diffuses over a distance that is proportional to a squareroot of time t. The shorter the duration in which power is applied toheating element 633 thus affects thinner interfacial layers of ice. Inone embodiment, the time t and energy Q applied to heating element633—to heat an interfacial layer of ice 630 from an ambient temperatureT to a melting point temperature T_(m) (melting point 638)—follows theequations discussed FIG. 1.

[0321]FIG. 49 shows one slider 640 to illustrate testing of frictionalchanges at the ice-object interface. Slider 640 includes acrylic slider644, force sensor 642, and heating element 646, such as Ti foil with athickness in a range of about 12.5 μm to 25 μm. Slider 640 employs aheating element 646 that melts an interfacial layer of ice 641 adjacentslider 640 by pulsating thermal energy to the layer, for example inaccordance with the equations of FIG. 1. Power may be applied to heatingelement 646 at terminals 645 and 647 such that heating element 646 meltsthe interfacial layer of ice 641. Once the interfacial layer of ice 641is melted, the interfacial layer of ice is allowed to refreeze due tocooler ambient temperature, providing a bond between ice 641 and slider644.

[0322] Force sensor 642 illustratively receives force information aboutthe force applied from slider 644 towards ice 641. Force sensor 642 mayrelay such information to a controller 643 for a determination of how toapply power to heating element 646. A power supply, such as thosedescribed herein, may then supply power to heating element 646 to meltthe interfacial layer of ice 641. Melting the interfacial layer of theice 641 modifies adhesion strength of ice 641 to slider 640 and changesa coefficient of friction between ice 641 and slider 644.

[0323]FIGS. 50 and 51 illustrate an application of one slider 650 in theform of ski 654, in accord with one embodiment. Slider 650 includesmetal heating elements 652, such as Ti foil, coupled with a ski surface651, which is in contact with snow 653. Heating elements 652 areconfigured for melting a layer of interfacial snow interfacing withsurface 651 by pulsating energy to the layer of snow 653, such as inaccordance with the equations of FIG. 1. Power is for example applied toheating elements 652 by one of several devices described herein. Oncethe interfacial layer of snow 653 is melted, it refreezes due to coolerambient temperature and provides a bond between snow 653 and surface651. The bond provides improved traction to snow 653 by modifying acoefficient of friction between the ice and slider 650.

[0324] Slider 650 may also include a binding 658, shown in FIG. 51.Switch 660 is located with binding 658 to control the manner in whichpower is delivered to heating elements 652. An example of switch 660 isa mechanical switch. Switch 660 may also include a manual switch, a skimotion switch, a pressure-activated switch, an accelerometer, remotecontrol switch, and/or a motion sensor; each such switch may be usedwith slider 650 to activate heating and refreezing of the interfaciallayer of ice to provide a desired coefficient of friction.

[0325] More particularly, FIG. 50 further shows the manner in whichheating elements 652 may be affixed to ski 654. In FIG. 51, a ski boot656 is inserted into binding 658. Ski boot 656 may be used to controlswitch 660, if desired, so that power is applied to heating elements652. Power may be supplied by power sources described herein. In oneexample, when boot 656 triggers switch 660, switch 660 conducts powerfrom a power supply to heating elements 652 to melt an interfacial layerof snow 653, thereby modifying a coefficient of friction between ski 654and snow 653.

[0326]FIG. 52 illustrates one slider 670 in the form of snowboard 674.Slider 670 includes heating elements 672 affixed to a bottom surface 675of snowboard 674; surface 675 is in contact with snow during operationof snowboard 674. Operative characteristics of slider 670 may be similarto those ski 654 of FIGS. 50 and 51. Heating elements 672 may also beinternal to snowboard 674, but in thermal communication with surface675, in accord with one embodiment.

[0327]FIG. 53 illustrates one slider 680 in the form of shoe 684. Slider680 includes metal heating elements 682, such as Ti foil, affixed toheel 688 and sole 686; heel 688 and sole 686 contact snow or ice when aperson walks on snow or on ice. Heating elements 682 may also beinternal to show 684 (or heel 686) so long as they are in thermalcommunication with the outer-most surface of heel 688. Heating elements682 may be made of a thin conductive film (e.g. TiN film, Cr film)sputtered on either a polymer substrate (e.g. Kapton, ABS) or on aceramic substrate (e.g. glass ceramic, zirconia ceramic). Power isapplied to heating elements 682 such that heating elements 682 melt aninterfacial layer of ice adjacent heel and/or sole 688, 686. Once theinterfacial layer of ice or snow is melted, it is allowed to refreezedue to ambient temperature, thereby providing a bond of ice or snow toheel and/or sole 688, 686. Power is applied to heating elements 652, forexample as described in connection with FIG. 1. In one embodiment,slider 680 employs a small battery 683 (e.g., a D-cell battery), as thepower supply. A switch, such as switch 48 of FIG. 4, connects the powerfrom the power supply to heating elements 682. In one example, when auser triggers the switch, the switch conducts power from battery 683 toheating elements 682, to melt an interfacial layer of the ice or snowand to modify a coefficient of friction between shoe 684 and the ice orsnow, assisting with traction of shoe 684.

[0328]FIG. 54 illustrates one slider 690 in the form of tire 692. Slider690 includes metal heating elements 694 embedded in tire 692. Power isapplied to heating elements 694 such that heating elements 694 melt aninterfacial layer of ice or snow 693. Once the interfacial layer of ice693 is melted, it refreezes due to ambient temperature and provides abond between ice/snow 693 and tire 692. Power may be applied to heatingelements 694 by one of several techniques discussed herein. In oneembodiment, slider 690 employs a car battery as its power supply.

[0329] In one example, heating elements 694 include thin metal wiresconfigured for receiving the power and converting that power intothermal energy, to melt the interfacial layer of ice/snow 693 in contactwith tire 692. Additionally, slider 690 may include a controller, suchas controller 78 of FIG. 6, to controllably apply that power accordingto the equations of FIG. 1. In one embodiment, a user activates a switch(e.g., similar to other embodiments described herein) so that the poweris applied to heating elements 694 when needed for additional tractionbetween tire 692 and a road surface covered with ice and snow 693. Inone example, when a user triggers a switch by depressing a preconfiguredbutton on a console in a car, the switch conducts power from the powersupply to heating elements 694 to melt an interfacial layer of the iceand snow 693, thereby modifying a coefficient of friction between tire692 and an ice and snow covering the road surface when the interfaciallayer refreezes and increases traction of tire 692 on the snow/ice 693.

[0330] Heating elements 694 thus may operate as a “pulse brakes” byproviding a heating pulse to the interface between tire 692 and snow/ice693. For example, when braking is needed, an interfacial layer of ice ismelted. When the pulse stops, melted spots on tire 692 typicallyre-freeze within a few milliseconds due to ambient temperature,providing strong bonds between tire 692 and ice/snow 693. These bondsassist in braking the motions of tire 692 relative to ice/snow 693. Inone embodiment, a Peltier element 695 is used to more rapidly cool themelted interfacial layer of ice.

[0331] An example of Peltier element 695 is a thermoelectric moduleconsisting of an array of Bismuth Telluride doped semiconductor pelletsof one type of charge carrier (e.g., positive or negative) for carryinga majority of current. Pairs of positive and negative pellets areconfigured so that they electrically connect in series, but thermallyconnect in parallel. Metalicized ceramic substrates may provide aplatform for the pellets. Thermoelectric modules may function singularlyor in groups with either series, parallel, or series-parallel electricalconnections.

[0332] When a DC voltage is applied to Peltier element 695, the positiveand negative charge carriers in the pellet array absorb heat energy fromone substrate surface and release it to an oppositely situatedsubstrate. The surface where heat energy is absorbed may decreasetemperature without moving parts, compressors, or gases. The oppositelysituated substrate, where heat energy is released, resultantly increasesin temperature.

[0333]FIG. 55 illustrates a test configuration of one slider 700, toillustrate how a slider affects friction to adjacent snow or ice. Slider700 includes a plurality of metal heating elements embedded in a region704 illustrating electrically conductive rubber of a tire. Power isapplied to heating elements 712 so as to melt an interfacial layer ofice 714. Once the interfacial layer of ice is melted, it refreezes dueto ambient temperature and provides a bond between ice 714 and slider700.

[0334] In one embodiment, heating element 712 is a thin metal wireconfigured for receiving the power and converting that power intothermal energy to melt the interfacial layer of ice 714 in contact withslider 700. A thin electrical insulator 706 about the heating elementmay surround heating element 712. As heating element 712 receives powerfrom power supply 702, the heating elements 712 convert the power intothermal energy through resistivity. The thermal energy is conducted(thermal radiation lines 710) to ice 714 and into a heated region 708,in which the interfacial layer of ice 714 is melted. The meltedinterfacial ice changes a coefficient of friction between the slider 700and ice 714 such that traction between slider 700 and ice 714 isincreased. The coefficient of friction changes due to melting andrefreezing as electrical power is respectively applied and removed toheating element 712. For example, a pulse of electrical power having aduration in accordance with Eq. 1.4 of FIG. 1 melts the interfaciallayer of ice 714 as it is converted to thermal energy by heating element712. As the pulse of electrical power subsides, region 708 is allowed torefreeze, due to cooler ambient temperature and non-melted ice 714. Thismelting and refreezing of ice 714 modifies the coefficient of frictionand improves traction and braking when, for example, slider 700 is anobject such as a tire or a ski.

[0335]FIG. 56 illustrates one slider 720 in the form of track 724 suchas used by a snowmobile. Slider 720 includes heating elements 722embedded in track 724. Power is applied to heating elements 722 suchthat heating elements 722 melt an interfacial layer of ice adjacenttrack 724. Once the interfacial layer of ice is melted and power is nolonger applied, the melted interfacial layer of water refreezes due toambient temperature and provides a bond of ice to track 724. In oneembodiment, slider 720 employs a battery as the power supply.Illustratively, track 724 is shown about track wheels 725. Heatingelements 722 may be in the form of thin metal wires or in the form ofthin metal foil that convert power into thermal energy to melt theinterfacial layer of ice in contact with track 724. A user may activatea switch as desired to apply power to heating elements 722, such as whenthe user determines a need for additional traction between track 724 anda terrain covered with ice and snow. When a user triggers the switch,the switch conducts power from a power supply (e.g., a snowmobilebattery) to heating elements 722 to melt the interfacial layer ofice/snow, thereby modifying a coefficient of friction between track 724and the snow, increasing traction of track 724 on the snow due tosubsequent refreezing.

[0336]FIG. 57 illustrates one slider 780 in the form of ski 782; ski 780is shown in greater detail in view 781. In the exemplary embodiment,slider 780 includes heating element 784 and may have operativecharacteristics similar ski 654 of FIGS. 50 and 51. Heating element 784(exaggerated in view 781 for purposes of illustration) may be formedfrom material such as Ti foil or abrasion-resistant conductive paints(e.g., nickel-based and silver-based paints), or a sputtered layer ofTiN. Heating element 784 is affixed to a surface of ski 782 (orotherwise arranged to thermally communicate with the surface) tocontinually contact snow and melt interfacial snow or ice, suchdescribed in connection with FIG. 1.

[0337] View 781 shows one manner in which heating element 784 may affixto ski 782. For example, view 781 shows an exploded view in whichheating element 784 is affixed to ski 782 via posts 783. Posts 783 aretypically formed as a metallic conductor to serve as electric busterminals, and also to shield heating element 784 from damage. Posts 783may be used to conduct power from a power supply to heating element 784to melt an interfacial layer of snow, thereby modifying a coefficient offriction between ski 782 and the snow.

[0338] In one embodiment, heating element 784 includes a protectivecoating 785 to guard against rock damage. Heating element 784, posts783, and substrate 786 may be replaceable. When heating element 784includes a conductive layer of paint, scratches may be repaired with atouch-up paint kit.

[0339]FIG. 58 illustrates one slider 800 in the form of tire 802, inaccord with one embodiment. Slider 800 includes heating unit 806 and anoptional air exhaust sub-system 804. Air exhaust sub-system 804 mayinclude a cold-air exhaust of an automobile air conditioner. Heatingunit 806 may include a heat lamp or other heating device to heat region805 of tire 802 with pulsed or continuous thermal energy. Slider 800 mayemploy a battery of a vehicle as the power supply.

[0340] In one embodiment, heating unit 806 includes or utilizes theexhaust of the cars's air conditioner or engine. In another embodiment,heating unit 806 includes or utilizes a water spray that generates finewater mist; the water mist covers a car tire with a thin water film,which freezes on contact with ice, thus providing strong bonds betweenthe tire and ice.

[0341] In another embodiment, heating unit 806 includes a hot cylindertouching the tire; the cylinder may rotate with the tire. The hotrotating cylinder may be heated by a car electrical system, by a car'sair conditioner, and/or by car exhaust gases.

[0342] In one operational example, heating unit 806 is configured toreceive power and to convert that power into thermal energy, to melt aninterfacial layer of ice 810 at region 807 in contact with tire 802. Asheating unit 806 receives power from the power supply, it converts thepower into thermal energy and forms heated region 805. Because of theshort duration of exposure to heat, typically only a thin layer of thetire rubber is heated. As tire 802 rotates, heated region 805 melts aninterfacial layer of ice 810 at region 807. As the tire continues torotate, the melted layer of ice refreezes at region 808 and changes acoefficient of friction between tire 802 and ice 810, at zone 809,thereby creating a bond between tire 802 and ice 810 such that tractionbetween tire 802 and ice 810 increases.

[0343] Because tire 802 has significant contact area with ice 810, therubber of tire 802 is usually re-cooled before it is again heated byheating unit 806; thus, additional cooling is normally not necessarywhen the ambient temperature is below the melting point of ice.Nonetheless, additional cooling may be used; for example, cold air fromthe car's air conditioner may be used to cool the tire via exhaustsub-system 804.

[0344] Since heating unit 806 can pulse thermal energy, the coefficientof friction may discretely change as a result of interfacial ice 810melting and refreezing as electrical power is respectively applied andremoved (e.g., tire 802 incrementally heats and cools as it rotates). Inone embodiment, heating unit 806 may include a heated metal brushpressed against rotating tire 802. The heat flux from the brush to thesurface 801 of tire 802 heats a thin layer of tire rubber to causesubsequent melting of the interfacial ice.

[0345] The mean power used by heating unit 806 typically depends onambient temperature and car velocity; but may be in a range of about 10watts to 100 watts. In certain extreme cases, it may be in the range ofabout 1 watt to 1000 watts. Also depending on these temperature andvelocity conditions, the duration in which the rubber of tire 802 isilluminated or heated by heating unit 806 is in a range of about 3 ms to100 ms, but may be from about 1 ms to 1 s in more extreme cases. Therefreezing time may be about the same as for a pulse deicer system, suchas those described in FIGS. 1-6 (e.g., typically in a range from about 1ms to 100 ms). These times may be adjusted so as to provide maximumtraction when most of the road-to-tire contact region is refrozen.

[0346]FIG. 59 illustrates a test configuration of one slider 820, inaccord with one embodiment. Slider 820 includes slider interface 825 andphotoflash lamp 826. Photoflash lamp 826 is configured to illuminateslider interface 825 with a pulse of light (e.g., a flash of light).Photoflash lamp 826 receives power from power supply 822 to melt aninterfacial layer of ice 821. Photoflash lamp 826 pulses light to a thinblackened layer 827 interfacing ice 821. A typical duration and energy,per pulse, of lamp 826 is about 1 ms to 10 ms, generating an energy ofabout 1J to 100J.

[0347] In one embodiment, a single flash from photoflash lamp 826 meltsthe interfacial layer of ice 821 as photoflash lamp 826 illuminatesslider interface 825. Slider interface 825 is typically transparent andconverts energy from flash into thermal energy as light impingesblackened layer 827. For example, light from lamp 826 (e.g., visiblelight or infrared light) is absorbed by layer 827 and converted intothermal energy. The converted thermal energy is then absorbed in aninterfacial layer of ice 821 adjacent slider 820. As the energy isabsorbed by the interfacial layer of ice 821, the layer melts. The layerthen refreezes due to ambient temperature to provide a bond betweenslider 824 and ice 821.

Coefficient of Friction Modification Analysis

[0348] Certain analyses are now described in which the coefficient offriction is modified at the ice-object interface or snow-objectinterface. These analyses may experimentally and graphically illustratemodification of a coefficient of friction.

[0349]FIG. 60 shows graph 830 illustrating an exemplary relationshipbetween coefficients of friction of certain sliders and voltage appliedto heating elements affixed to the sliders, in accord with oneembodiment. An electric circuit such as shown in FIG. 2 was used tocharge a 2.35 mF capacitor. The capacitor was then discharged throughthe heating element. In FIG. 60, Y-axis 831 represents frictional forceand X-axis 832 represents voltage. Graph 830 distinguishes between twosimilar sliders, each with a heating element (one heating elementincludes Ti foil of about 12.5 μm thickness and the other heatingelement includes Ti foil of about 25 μm thickness). At about 50V ofpower applied to the heating elements, the coefficients of frictionbetween the sliders and the snow changes, as shown. At about 100V, thecoefficients of friction of the sliders to the snow begin todifferentiate from one another. Accordingly, the thickness of theheating element material is substantially independent of voltage untilabout 100V, which may affect design considerations.

[0350]FIG. 61 shows graph 840 illustrating an exemplary relationshipbetween static force of certain sliders and normal pressure of thesliders exerted on snow, in accord with one embodiment. In FIG. 61,Y-axis 841 represents static force and X-axis 842 represents normalpressure. Graph 840 distinguishes between two similar sliders, each witha heating element (one heating element includes Ti foil of about 12.5 μmthickness and the other heating element includes Ti foil of about 25 μmthickness). The two graphs below show static force of friction for thesame sliders as measured without heating pulses applied. Otherexperimental details, such as DC voltage (90 V), temperature (−11° C.),and the capacitor used in the circuit of FIG. 2, are shown in the graphinsert.

[0351]FIG. 62 shows graph 850 illustrating an exemplary relationshipbetween coefficients of friction of certain sliders and the voltageapplied to an affixed heating element, in accord with one embodiment. InFIG. 62, Y-axis 853 represents frictional force and X-axis 852represents voltage. Graph 850 distinguishes between two similar sliders,each with a heating element (one heating element includes Ti foil ofabout 12.5 μm thickness and the other heating element includes Ti foilof about 25 μm thickness). Each slider has an average curve asdetermined by a range of coefficients of friction associated with aparticular applied voltage. For example, a slider with a heating elementhaving Ti foil with a 25 μm thickness has a coefficient of friction thatvaries in a range of about 4.9N to 6N (point 851). FIG. 62 demonstratesthat the pulse brake works well even when ambient temperature is veryclose to the melting point (−2° C.); good braking force is achieved evenat −0.5° C.

[0352]FIG. 63 shows graph 860 illustrating an exemplary relationshipbetween coefficients of friction of one slider and the time duringsliding at constant velocity of 3.5 mm/s. In FIG. 63, Y-axis 863represents frictional force and X-axis 864 represents time. Four shortpulses of heating power were applied during the experiment, during whichthe slider moved at a velocity of about 3.5 mm/s. A 1.36 mF capacitordischarged current to the heating element at about 110V in four pulses861. The duration of the heating pulses were about 2.5 ms. A heatingelement affixed to the slider received power from the power supply for alimited duration (as a pulse of power), for example in accord with theequations of FIG. 1. The heating element converted that power intothermal energy and applied the thermal energy to the surface-to-iceinterface. The heating element melted an interfacial layer of snow orice adjacent to the slider. Melting the interfacial layer modifies theadhesion of the snow at the slider's surface and changes the coefficientof friction between the slider and the snow or ice. During each pulse861, the coefficient of friction changes. The changing coefficient offriction between the slider and the snow causes the slider to resistsliding, thus increasing the friction force. That can be seen in FIG. 63as the sharp peaks in the friction force. Changing the pulse energy andintervals between pulses, one can adjust an average friction force to adesirable magnitude. Those skilled in the art understand that such anadjustable brake may couple with a velocity-measuring system tofacilitate making the ski a “cruise-control” system: a skier can preseta desirable maximum speed for himself or his children to have safeskiing.

[0353]FIG. 64 shows graph 870 illustrating another exemplaryrelationship between coefficients of friction of one slider and voltageapplied to an affixed heating element, in accord with one embodiment. InFIG. 64, Y-axis 871 represents frictional force and X-axis 872represents voltage. In this embodiment, the voltage was varied todetermine coefficients of friction as dependent upon power. At about 50Vof power applied to the heating elements, the coefficient of frictionchanged. At about 90V, the coefficient of friction of the slider to thesnow saturates and then remains almost constant until about 110V.Accordingly, a voltage between 90V and 110V may provide an increase inthe coefficient of friction that is substantially independent of voltagebetween the 90V and 110V. This information is useful when choosing apower supply for a slider design.

[0354]FIGS. 65 and 66 show graphs illustrating thermal energy Q andcooling time t_(cool) of one slider. In FIG. 65, Y-axis 881 representsheat diffusion length in snow L_(D) and X-axis 882 represents time. InFIG. 66, Y-axis 891 represents thermal energy and X-axis 892 representsresistance of a heater. In the example, during a first 10 millisecondsof heating the heat penetrates snow only to depth of thirty-six microns.Such a thin snow layer has a small heat capacity, requiring littleenergy to heat it to the melting point (i.e. 273K). Table 65-1 belowcalculates a total energy Q(Δ,R) used to melt a ten-micron thick layerof ice and to heat the interfacial snow and ski material by Δ degrees C.When heating power does not depend on T, the result is shown in Table65-1: TABLE 65-1 W := 10⁴, 2 · 10⁴ . . . 10⁶ λ_(ski) := 0.2 ρ_(ski) :=1000 C_(ski) = 1.54 × 10³ ρ_(snow) := 300 C_(snow) := 2.2 · 10³ λ_(snow):= 0.2 $D_{snow}:=\frac{\lambda_{snow}}{\rho_{snow} \cdot C_{snow}}$

s R := 0.1, 0.2 . . . 10 ohm C := 10⁻⁴, 2 · 10⁻⁴ . . . 2 · 10⁻² F t(R,C) := R · C $D_{ski}:=\frac{\lambda_{ski}}{\rho_{ski} \cdot C_{ski}}$

Δ := 0.01, 0.02 . . . 10 t := 0, 10⁻⁴ . . . 10⁻¹

[0355] As illustrated in FIGS. 65 and 66, the heat diffusion lengthL_(D) (e.g., plot 880, FIGS. 65), is:${{L_{D}(t)}\text{:}} = \sqrt{D_{snow} \cdot t}$

L _(D)(10⁻²)=5.505×10⁻⁵

L _(D)(1)=5.505×10⁻⁴

L _(D)(0.1)=1.741×10 ⁻⁴

L _(D)(0.01)=5.505×10⁻⁵

V:=100

S:=0.0025

[0356] ${{W(R)}\text{:}} = \frac{V^{2}}{2 \cdot R \cdot S}$

d _(heater):=1.25·10⁻⁵

C_(heater):=523

ρ_(heater):=4.5·10³

l _(melt):=1×10⁻⁵

q _(latent):=3.33·10⁵

[0357]$Q = {{{\frac{{\pi\Delta}^{2}S}{4{W(R)}}\left\lbrack {\sqrt{\rho_{snow}c_{snow}\lambda_{snow}} + \sqrt{\rho_{ski}c_{ski}\lambda_{ski}}} \right\rbrack}^{2} + {d_{i} \cdot q_{i} \cdot \rho_{i}} + {d_{heater}C_{heater}\rho_{he}{C\left( {\Delta,R} \right)}\text{:}}} = \frac{2 \cdot {Q\left( {\Delta,R} \right)}}{V^{2}}}$

C(20,2.5)=8.464×10⁻⁴

Δ:=20

d _(heater) ·S·ρ _(heater) ·Δ·C _(heater)=1.471

l _(melt·ρ) _(snow) ·S·q _(latent)=2.498

[0358] where S is heater area, T_(m) is melting temperature, T isambient temperature, λ is a thermal conductivity coefficient, ρ is thematerial density, and C is the material heat capacity (subscript “ice”denotes ice and/or snow, subscript “ski” denotes substrate material,such as a ski or a snowboard, subscript “heater” denotes a heatingelement), Q is thermal energy, D is a heat diffusivity coefficient, Δdenotes temperature change, t is time, V is voltage, d is thickness, Ris resistance, W is a power per square meter, l_(melt) is thickness ofmelted layer, and q is latent heat of melting. Accordingly, for veryshort pulses, nearly all thermal energy Q is used to melt a thin layerof snow (plot 890, FIG. 66); snow and ski heat capacitance contributeslittle to Q. A calculation of refreezing time for the melted layer isshown by the following Table 65-2: TABLE 65-2 λ_(ski) := 0.5 λ_(snow) :=0.5${t_{cool}\left( {\Delta,R} \right)}:=\left\lbrack \frac{2{Q\left( {\Delta,R} \right)}}{\Delta \cdot S \cdot \left( {\sqrt{\lambda_{snow} \cdot \rho_{snow} \cdot C_{snow}} + \sqrt{\lambda_{ski} \cdot \rho_{ski} \cdot C_{ski}}} \right)} \right\rbrack^{2}$

t_(cool)(20, 1) = 0.013″s″

[0359] Table 65-3 illustrates typical capacities of common batteriesused as power supplies in pulse brake applications. For example, a pairof small AA batteries may be used in a pulse brake application by across-country skier for about a one-hour run. TABLE 65-3 Battery sizeType Voltage A · h watt · hour 1. AA, Duracell ordinary 1.5 2.85 4.275two of them 3 5.7 8.55 2. C, Duracell ordinary 1.5 7.8 11.7 two of them3 15.6 23.4 3. D, Duracell ordinary 1.5 15 22.5 two of them 3 30 45 4.D, Varta ordinary 1.5 16.5 24.75 two of them 3 33 49.5 5. 9v, Duracellordinary 9 0.58 5.22 two of them 18 1.16 10.44 4 of them 36 20.88 --noconverter is needed-- 6. D-Type Li-ion TL2300/S D, Li recharge- 3.6 16.559.4 two of them able 7.2 33 ($20.65) 118.4 ($41.30) 7. DD Li-ionTL5137/TDD, Li recharge- 3.6 35 126 able ($48.93) 8. AA Li-ionTL5104/PT2 AA, Li recharge- 3.6 2.1 7.56 able 9. C Li-ion TL2200/SC, Li,recharge- 3.6 7.2 25.92 7200 mAh able 7.2 14.4 ($16.73) two of them 52

[0360]FIG. 67 shows one analysis of one slider 900 illustratingfriction-enhancement for an embodiment wherein the slider forms a tire902. Slider 900 shows tire 902 with differing thermal zones in supportof this analysis: φ₀ is a heated zone; φ₁ is an air-cooled zone; φ₂ is amelting zone; φ₃ is a refreezing zone; φ₄ is a bonding zone; ω₀ isangular velocity of the tire; ν₀ is linear velocity of the car; R is theradius of tire 902; and A is the width of tire 902. Assuming that heatedzone φ₀ is uniformly heated with total power w′, then the power densityper square meter may conform to the following: $\begin{matrix}{w = {\frac{w^{\prime}}{R \cdot \phi_{0} \cdot A}.}} & \left. \text{(Eq.~~67-1} \right)\end{matrix}$

[0361] Each point inside the heated zone φ₀ may be “surface-heated” fortime t as follows: $\begin{matrix}{t = {\frac{\phi_{0}}{\omega} = {\frac{\phi_{0}R}{\upsilon}.}}} & \text{(Eq.~~67-2)}\end{matrix}$

[0362] For example, at$t = {\frac{0.1\quad {m \cdot s}}{{3 \cdot 10^{1\quad}}m} \approx {{3.3 \cdot 10^{- 3}}\quad s,}}$

[0363] an φ₀R=0.1 m,$\nu_{0} = {30\frac{m}{s}\left( {108\frac{km}{h}} \right)}$

[0364] and the heated zone φ₀ acquires an energy density of:$\begin{matrix}{Q = {{t \cdot w} = {\frac{w^{\prime} \cdot \phi_{0} \cdot R}{{R \cdot \phi_{0}}{A \cdot \upsilon_{0}}} = {\frac{w^{\prime}}{A \cdot \upsilon_{0}}.}}}} & \left( {{{Eq}.\quad 67}\text{-}3} \right)\end{matrix}$

[0365] Estimating a minimum Q and assuming 10 μm thickness of melted iceyields the following:

Q=d·q·ρ _(i),   (Eq. 67-4)

[0366] where

[0367] d is melted layer thickness in φ₂-zone, ρ_(i) is ice density, andq is the ice latent heat of fusion. Accordingly, $\begin{matrix}{{d \cdot q \cdot \rho_{i}} = {\frac{w^{\prime}}{A \cdot \upsilon_{0}},}} & \left( {{{Eq}.\quad 67}\text{-}5} \right)\end{matrix}$

[0368] and, therefore,

w′=A·ν ₀ ·d·q·ρ _(i).

[0369] An estimate of the re-freeze area which would increase thefriction coefficient to μ=0.5 is now determined. For example, at anormal pressure of 2·10⁵ Pa, the friction force per square metercorresponding to μ=0.5 is 10⁵ Pa. For an ice/rubber interface, adhesionshear strength is about 1 Mpa; thus only about 10% of the ice/tirecontact area may need refreezing (e.g., refreezing zone φ₃) to provideμ=0.5. When a melted layer of ice has a thickness of about 3.3 μm, thepower requirement is about 500 watts for a velocity ν₀ equal to about$108{\frac{km}{h}.}$

[0370] For a velocity ν₀ of about $7.2\frac{km}{h}$

[0371] at the same thickness, the power requirement is only about 33watts.

[0372] At a velocity ν₀ of 20 km/h, every point on the tire surface maybe in contact with the ice for about$t = {\frac{{2 \cdot 10^{- 1}}m}{6\quad m\text{/}s} = {30\quad m\quad {\sec.}}}$

[0373] This time is available for melting and refreezing actions, and islong enough to accomplish such actions.

[0374]FIGS. 68 and 69 illustrate experimental results in which icefriction was reduced by either application of HF-power, as in FIG. 68,or by application of low-energy heating pulses, as in FIG. 69. In FIG.68, Y-axis 915 represents frictional force and X-axis 914 representstime in seconds. For example, FIG. 68 shows a frictional force N versustime for the slider in motion on ice with an ambient temperature T ofabout −5° C., a normal pressure P of about 42 kPa, and a slidingvelocity v of about 1 cm/s. In this embodiment, the system modifying thefriction includes an interdigitated circuit attached to a base of theslider that interfaces with ice. The interdigitated circuit includes acopper clad Kapton polyimid film. The interdigitated circuit alsoincludes copper electrodes having an inter electrode spacing of about 75μm. A power supply provided HF AC voltage of about 30V rms at about 20kHz to the electrodes. The electrodes generated heat in ice of about 100watts/m² density. When the slider moves at a velocity of about$1\frac{cm}{s}$

[0375] and the power is applied to the electrodes, the friction force islower by about 40%. For example, the power supply provided the HF-powerto the electrodes at time point 910 (e.g., about time t equal to 10 s).The electrodes converted the power into thermal energy which diffused inthe direction of the ice. The slider begins sliding at time point 912(e.g., about time t equal to 13 s). In this embodiment, the HF-power isshut down at time point 911 (e.g., about time t equal to 28 s). Withoutthe HF-power the ice friction rises from 4 N to 7 N. The latter is abackground ice friction force with no power applied to the slider, whichstopped at time point 913 (e.g., about time t equal to 33 s).

[0376] In this embodiment, the continuous HF-power supply increases theice temperature, thus decreasing ice friction without generating icemelt and, thereby modifying the coefficient of friction.

[0377]FIG. 69 shows a frictional force N versus time for the slider inmotion on snow with an ambient temperature T of about −10° C., a normalpressure P of about 215 kPa, and a sliding velocity ν of about 3 mm/s.In FIG. 69, Y-axis 925 represents frictional force and X-axis 926represents time in seconds. In this embodiment, the system modifying thefriction includes a thin titanium-foil heater. Short heating pulses ofDC power are applied to the heater at time moments 922 and 923 causingdecrease in snow friction, as opposed to the braking effect by the samesystem described earlier. The main difference of this experiment is thepulse braking; as shown in FIG. 69, the magnitude of heating energy isnot sufficient to melt snow. Without a melted layer, refreezing does notoccur and there is no braking action. Nevertheless, since the heaterwarms snow, the friction decreases. In the experiment of FIG. 69, thesnow surface is heated by the pulses from −10° C. to about −1° C. Theslider experiences a rapid increase in static friction between the iceand the slider at time point 921 (e.g., about time t equal to 31 s). Thepower supply provides pulse power at time points 922 and 923 (time tequal to 38 s and 42 s, respectively) to the electrodes. In thisembodiment, the slider stops at time point 924, when time t equals 50 s.

[0378] In some embodiments, the electrodes of the interdigitated circuitare made of hard conductive materials, such as titanium nitride,zirconium oxide (e.g., zirconia) doped with other oxides (e.g., ittriumoxide), and titanium and stainless steel foils with TiN coatings, toincrease abrasion resistance of the circuit. Other embodiments mayprovide electrode protection through coatings of protective films, suchas alumina.

[0379] Since certain changes may be made in the above methods andsystems without departing from the scope, it is intended that all mattercontained in the above description or shown in the accompanying drawingsbe interpreted as illustrative and not in a limiting sense. It is alsoto be understood that the following claims are to cover all generic andspecific features described herein, and all statements of the scopewhich, as a matter of language, might be said to fall there between.

What is claimed is:
 1. A method of thermally modifying an interfacebetween ice and an object, comprising the steps of: applying heatingenergy to the interface to melt an interfacial layer of ice; andlimiting duration of the step of applying heating energy to theinterface such that the heating energy has a heat diffusion distancewithin the ice that extends no more than through the thickness of theinterfacial layer of ice.
 2. The method of claim 1, wherein the step ofapplying heating energy comprises the step of applying power at theinterface with a magnitude that is at least about inverse proportionalto a magnitude of energy used to melt the interfacial layer of ice. 3.The method of claim 2, wherein the step of limiting duration comprisesthe step of limiting duration of the step of applying power at theinterface such that the duration is at least about inverse proportionalto a square of the magnitude of the power.
 4. The method of claim 1,wherein the step of applying heating energy comprises the step ofapplying power to the interface with a magnitude that is substantiallyinverse proportion to a magnitude of energy used to melt the interfacialice, and wherein the step of limiting duration comprises the step oflimiting the duration so that the duration is substantially inverseproportion to a square of the magnitude of the power.
 5. The method ofclaim 1, further comprising the step of facilitating refreezing of theinterfacial layer of the ice to affect a coefficient of friction betweenthe object and the ice.
 6. The method of claim 5, the step offacilitating comprising one or more of the following steps: (1) waitingfor refreezing after the step of limiting duration; (2) blowing cold airat the interface; and (3) misting water at the interface.
 7. The methodof claim 1, the object selected from the group of an aircraft structure,a windshield, a mirror, a headlight, a power line, a ski lift structure,a rotor surface of a windmill, a rotor surface of a helicopter, a roof,a deck, a building structure, a road, a bridge structure, a freezerstructure, an antenna, a satellite, a railroad structure, a tunnelstructure, a cable, a road sign, a snowshoe, a ski, a snowboard, askate, and a shoe.
 8. The method of claim 1, wherein the step ofapplying heating energy to the interface comprises applying heatingenergy to the interface to melt an interfacial layer of ice having athickness that is less than about five centimeters.
 9. The method ofclaim 1, wherein the step of applying heating energy to the interfacecomprises applying heating energy to the interface to melt aninterfacial layer of ice having a thickness that is less than about onemillimeter.
 10. The method of claim 1, wherein the step of applyingheating energy to the interface comprises applying heating energy to theinterface to melt an interfacial layer of ice having a thickness that isbetween about one micron and one millimeter.
 11. The method of claim 1,wherein the step of limiting duration of the step of applying heatingenergy to the interface comprises the step of applying heating energy tothe interface for a maximum of 100 s.
 12. The method of claim 1, whereinthe step of applying heating energy to the interface comprises the stepof applying power to a heating element in thermal communication with theinterface.
 13. The method of claim 12, wheren the step of applyingheating energy to the interface comprises the step of applying power toa heating element within the object.
 14. The method of claim 12, whereinthe step of applying heating energy to the interface comprises the stepof applying power to a heating element at a surface of the object and incontact with the interface.
 15. The method of claim 12, wherein the stepof applying heating energy to the interface comprises the step ofelectrically resisting the power with the heating element.
 16. Themethod of claim 12, wherein the step of limiting duration comprises thestep of controlling duration of the step of applying power according tothe following relationship:${t = {\frac{{\pi \left( {T_{m} - T} \right)}^{2}}{4W^{2}}\left\lbrack {\sqrt{\rho_{i}c_{i}\lambda_{i}} + \sqrt{\rho_{s}c_{s}\lambda_{s}}}\quad \right\rbrack}^{2}},$

where it is the duration, T_(m) is an ice melting temperature, T isambient temperature, λ_(i) is a thermal conductivity coefficient of theice, ρ_(i) is a material density of the ice, c_(i) is a heat capacity ofthe ice, λ_(s) is a thermal conductivity coefficient of one or both ofthe object and the heating element, ρ_(s) is a material density of oneor both of the object and the heating element, c_(s) is material heatcapacity of one or both of the object and the heating element, and W isthe power.
 17. The method of claim 12, wherein the step of applyingpower comprises the step of controlling energy according to thefollowing relationship:$Q = {{\frac{{\pi \left( {T_{m}\quad - \quad T} \right)}^{2}}{4\quad W}\left\lbrack \quad {\sqrt{\rho_{i}\quad c_{i}\quad \lambda_{i}}\quad + \quad \sqrt{\rho_{s}\quad c_{s}\quad \lambda_{s}}} \right\rbrack}^{2},}$

where Q is energy that thermally melts the interfacial ice, T_(m) is atemperature to melt the interfacial ice, T is ambient temperature, λ_(i)is a thermal conductivity coefficient of the ice, ρ_(i) is a materialdensity of the ice, c_(i) is material heat capacity of the ice, λ_(s) isa thermal conductivity coefficient of one or both of the heating elementand the object, ρ_(s) is a material density of one or both of theheating element and the object, c_(s) is material heat capacity of oneor both of the heating element and the object, and W is the power. 18.The method of claim 12, wherein the step of applying power comprises thestep of controlling energy according to the following relationship:${Q = {{\frac{{\pi \left( {T_{m} - T} \right)}^{2}}{4W}\left\lbrack {\sqrt{\rho_{i}c_{i}\lambda_{i}} + \sqrt{\rho_{s}c_{s}\lambda_{s}}} \right\rbrack}^{2} + {d_{i} \cdot q_{i} \cdot \rho_{i}} + {d_{heater}C_{heater}{\rho_{heater}\left( {T_{m} - T} \right)}}}},$

where Q is the energy, T_(m) is a temperature for melting theinterfacial ice, T is ambient temperature, λ_(i) is a thermalconductivity coefficient of the ice, ρ_(i) is a material density of theice, c_(i) is material heat capacity of the ice, λ_(s) is a thermalconductivity coefficient of one or both of the heating element and theobject, ρ_(s) is a material density of one or both of the heatingelement and the object, c_(s) is material heat capacity of one or bothof the heating element and the object, d_(i) is a thickness of aninterfacial layer of ice, ρ_(i) is ice density, q_(i) is ice latent heatof fusion, W is the power, and C_(heater) and ρ_(heater) are specificheat capacity and density, respectively, of the heating element.
 19. Themethod of claim 1, further comprising the step of repeating the steps ofapplying and limiting in a periodic manner to generate a desiredcoefficient of friction between the object and the ice.
 20. The methodof claim 1, the step of limiting duration comprising the step oflimiting the duration to between about 1 ms to 10 s.
 21. The method ofclaim 1, further comprising reapplying power at the interface after theinterfacial layer refreezes to selectively control a coefficient offriction between the ice and the object while the object moves over theice.
 22. The method of claim 1, the ice comprising snow.
 23. The methodof claim 1, the object comprising a slider.
 24. The method of claim 23,the slider comprising one of a shoe, a snowboard, and a ski.
 25. Amethod for controlling a coefficient of friction between an object andice, comprising the steps of: (1) pulsing power to an interface betweenthe object and the ice to melt an interfacial layer of ice at theinterface and decrease the coefficient of friction; (2) facilitatingrefreezing of the interfacial ice at the interface to increase thecoefficient of friction; and (3) repeating steps (1) and (2) in acontrollable manner to control an average coefficient of frictionbetween the object and the ice.
 26. The method of claim 25, the step offacilitating refreezing comprising the step of moving the object overthe ice to decrease temeprature of the object.
 27. The method of claim25, the step of pulsing power comprising the steps of blowing first aironto the object, the first air having a temperature above freezing, andmoving the object in contact with the ice.
 28. The method of claim 27,the object comprising a tire of a vehicle.
 29. The method of claim 27,the step of faciliting refreezing comprising the step of blowing secondair onto the object, the second air having a temperature less than thetemperature of the first air.
 30. A slider having a surface intended tointerface with ice or snow, comprising: a power supply for generatingpower; a heating element configured to convert the power to heat at thesurface, the heat being sufficient to melt an interfacial layer of iceat the interface; a controller for controlling delivery of power to theheating element to control a coefficient of friction between the sliderand the ice or snow.
 31. The slider of claim 30, wherein the slidertakes the form of one of a shoe, a snowboard, a ski, and a snowshoe. 32.The slider of claim 30, the power supply comprising a battery.
 33. Theslider of claim 30, the slider being in a form of one of a ski, a skateand a snowboard, wherein the controller is responsive to user commandsto modulate power applied to the surface such that speed of the slideris controllable.
 34. A system for thermally modifying an ice-to-objectinterface, comprising: a power supply for generating power; a heatingelement coupled to the power supply to convert the power into heat atthe interface; and a controller coupled to the power supply to limit aduration in which power is applied to the heating element such that onlyan interfacial layer of ice melts at the interface.
 35. The system ofclaim 34, the interfacial layer having a thickness less than about fivecentimeters.
 36. The system of claim 34, the interfacial layer having athickness between about one micron and one millimeter.
 37. The system ofclaim 34, the power supply configured for generating the power with amagnitude that is substantially inverse proportion to a magnitude ofenergy which melts the interfacial ice; the controller configured tolimit the duration such that the duration has a substantially inverseproportion to a square of the magnitude of the power.
 38. The system ofclaim 34, further comprising a sensor coupled with the controller fordetecting temperature of the interface and for generating a feedbacksignal representative of the temperature to the controller.
 39. Thesystem of claim 34, the power supply comprising at least one of abattery, a capacitor, a flywheel, high-voltage power supply.
 40. Thesystem of claim 39, the capacitor comprising at least one of asupercapacitor, electrolytic capacitor, and an ultracapacitor.
 41. Thesystem of claim 34, the heating element comprising a thin film ofconductive material that transfers the heat from the heating element tothe interface to change a coefficient of friction between the object andthe ice.
 42. The system of claim 34, the heating element comprising asemiconductor material that converts the power into heat at theinterface to change a coefficient of friction between the object and theice.
 43. The system of claim 34, further comprising a switch coupled tothe controller for receiving a control signal from the controller tolimit the duration in which the power is applied to the heating element.44. The system of claim 34, the power supply, heating element andcontroller being configued with an object that forms the ice-to-objectinterface, the object being selected from the group consistingessentially of an aircraft, a windshield, a mirror, a headlight, a powerline, a ski lift structure, a rotor structure of a windmill, a rotorstructure of a helicopter, a roof, a deck, a building structure, a road,a bridge structure, a freezer structure, an antenna, a railroadstructure, a tunnel structure, a cable, a train structure, a shipstructure, a drilling platform, an icemaker structure, and a road sign.45. A method for heating an object to a temperature T, comprising thesteps of: applying power W to the object in a magnitude approximatelyinversely proportional to the energy sufficient to raise the temperatureof the object to temperature T; and controlling time of the appliedpower W in a duration inversely proportional to W².
 46. A method forcooling an object to a temperature T, comprising the steps of: removingpower W from the object in a magnitude inversely proportional to theenergy sufficient to cool the temperature of the object to temperatureT; and controlling time of the power W in a duration inverselyproportional to W².
 47. A windshield deicer, comprising: a windshield;and a substantially transparent heating element disposed with thewindshield that generates heat in response to applied power in amagnitude sufficient to melt an interfacial layer of ice on thewindshield.
 48. The windshield deicer of claim 47, the heating elementbeing selected from visually transparent semiconductor material havingan electron gap larger than about 3 eV.
 49. The windshield deicer ofclaim 48, the material comprising one of ZnO, ZnS, and mixtures thereof.The windshield deicer of claim 47, the heating element being selectedfrom transparent conductor material. The windshield deicer of claim 50,the material comprising one of indium tin oxide (ITO), tin oxide, thinmetal films, and mixtures thereof.
 52. The windshield deicer of claim47, further comprising a protective coating on the heating element. 53.The windshield deicer of claim 47, further comprising a power supply forgenerating the power.
 54. The windshield deicer of claim 51, the powersupply comprising a vehicle battery.
 55. The windshield deicer of claim47, further comprising a controller for limiting duration of the appliedpower such that a heat diffusion distance into ice is less than about athickness of the interfacial layer.
 56. The windshield deicer of claim53, the thickness being between about one micron and one millimeter. 57.The windshield deicer of claim 53, the heating element configured into aplurality of heating elements forming a plurality of segmented regions,the controller configured for applying the power to each of theplurality of heating elements to de-ice segments of the windshield. 58.A method of modifying friction between an object and ice/snow,comprising the steps of: applying a first pulse of thermal energy to aninterface between the object and the ice/snow, the first pulse beingsufficient to melt an interfacial layer of ice/snow adjacent the object;and refreezing water forming the interfacial layer to form a first bondbetween the object and the ice/snow.
 59. A method of claim 58, furthercomprising the steps of applying a second pulse of thermal energy, afterthe step of refreezing, to re-melt at least part of the interfaciallayer and refreezing the re-melted interfacial layer to form a secondbond between the object and the ice/snow.
 60. A method of claim 58, thestep of applying comprising pressing a hot cylinder against an object inthe form of a car tire.
 61. A method of claim 60, further comprisingelectrically heating the hot cylinder.
 62. A method of claim 60, thestep of refreezing comprising pressing a cold cylinder against the cartire.
 63. A method of claim 58, the step of refreezing comprisingutilizing car air conditioning.
 64. A method of claim 58, the step ofapplying comprising applying pulse power to a metal heater in thermalcommunication with the interface.
 65. A method of claim 64, the heaterbeing formed of a material with low heat capacity.
 66. A method of claim64, further comprising the steps of discharging a capacitor to theheater.
 67. A method of claim 66, further comprising the step ofcharging the capacitor with a power supply.
 68. A method of claim 67,further comprising utilizing a switch to charge and respectivelydischarge the capacitor.
 69. A method of claim 58, the step of applyingcomprising coupling a hot plate through one or more holes to a sliderbase adjacent the ice/snow and part of the object.
 70. A method of claim58, the step of applying comprising applying electrical pulse power to aheater coupled with a base of an object in the form of a ski.
 71. Amethod of claim 70, further comprising activating braking action of theski by one of a manual switch, a ski motion switch, an accelerometer, apressure-activated switch, and a motion sensor.
 72. A method of claim58, the step of applying comprising applying pulsed electrical energy toone or more heating elements in the base of an object in the form of asnowboard.
 73. A method of claim 58, further comprising utilizing aportable battery as a source of energy for the step of applying a pulseof thermal energy.
 74. A method of claim 58, the step of applyingcomprising utilizing a pulse action lamp to heat the object.
 75. Amethod of claim 74, the object in the form of a rotating tire, lightfrom the lamp temporarily heating respective zones of the tire whichmelt the layer, rotation of the tire refreezing the layer.
 76. A methodof claim 58, the step of applying comprising utilizing a metal brushagainst an object in the form of a car tire.
 77. A method of claim 58,the step of applying comprising utilizing a photoflash lamp illuminatingthrough the object to the layer.
 78. A method of claim 58, the step ofapplying comprising utilizing a car exhaust to heat the hot cylinder.79. A method of claim 58, the step of refreezing comprising utilizing acar air conditioner.
 80. A method of claim 58, the step of refreezingcomprising utilizing an electric Peltier's element.
 81. A method ofheating an object to a desired temperature, comprising steps of: storingthermal energy insulated from the object and in a magnitude at leastsufficient to heat the object to the desired temperature; and adjustingone or both of physical and thermal properties of an interface betweenthe thermal energy and the object to transfer at least part of thethermal energy to the object.
 82. The method of claim 81, wherein themethod further comprises a step of transferring the energy to aninterfacial layer of ice to disrupt adhesion of ice to a surface of theobject, the desired temperature being zero degrees Celcius or higher.83. The method of claim 81, wherein the step of transferring comprisesdisrupting the adhesion of ice to the surface of at least one of anaircraft, an aircraft wing, an automobile windshield, a boat, a road, abridge, a sidewalk, a freezer, a refrigerator, an icemaker, a ship, atrain, a drilling platform, a building, a runway, and a window.
 84. Themethod of claim 81, the step of adjusting comprising transferring thethermal energy from a first surface of a membrane to a second surface ofthe membrane through deflation of the membrane.
 85. The method of claim81, the step of adjusting comprising periodically pulsing the interfaceto provide periodic heating of the object.
 86. The method of claim 85,wherein the step of periodically pulsing comprises a step ofperiodically moving components of a heating element to modify a heattransfer rate between a heat storage and the object.
 87. The method ofclaim 86, the components comprising a plurality of grooved insulatingelements.
 88. The method of claim 87, the step of storing comprisingheating one of a liquid and/or gas, the step of adjusting comprisingflowing the liquid or gas adjacent to the object such that thermalenergy from the liquid or gas transfers to the object, to heat theobject.